The Garou from White Wolf's Werewolf: The Apocalypse. Each of the Changing Breeds was created to serve Gaia, and the Garou served as her warriors, protecting her from all things "of the Wyrm." Unfortunately, the Garou thought they were in a position to tell the other Breeds how to do their jobs. On top of that, Gaia never told them about some of the Breeds, or even some of the other tribes of Garou. Genocide ensued. By the time the game begins, three Changing Breeds and one whole tribe of Garou have been driven to extinction, the other Changing-Breeds are mostly in hiding, and the Garou have finally come to realize just how much they shot themselves in the foot. Furthermore, while most of the tribes have valid motivations, some of them go much, much too far. Don't ask about Red Talon "Christmas trees."
To be fair to the Garou, they were doomed from the start since the Triat, the uber-gods who embody the cycle of Creation, Stasis, and Destruction, are all completely screwed up. Instead of the Wyld creating, the Weaver defining, and the Wyrm destroying (the natural order of things), the Weaver went crazy and captured the Wyrm in a web, making the Wyrm go completely out of control. The Wyld, the only one of the Triat that might be on the Garou's side, is still attending to its duties; trouble is, due to its nature, it's not exactly sane. The Garou have several other gods on their side, including Gaia, the spirit of the planet itself, and Luna, the spirit of the moon, but it's unlikely to be enough. They're fighting to protect a system that is doomed from the get-go.
The Technocracy from Mage: The Ascension is an organization dedicated to making the world safe and predictable for sleepers. Unfortunately, their plan includes genocidally exterminating all supernatural creatures they find, even those who are also trying to protect the helpless and innocent, as well as attempting to monopolize scientific research and advancement, and lying to the sleepers about the true nature of reality.
But from their point of view the supernatural creatures include parasitic bloodsuckers that seek to control all society in their games, genocidal furry monsters that want to return humanity to Stone Age population levels, and dream parasites that latch onto people and use them as hosts.
This essentially defines the Banishers. They believe the supernatural, especially Mages, to be inherently evil (not necessarily without reason). Their solution is to attempt to wipe out every single supernatural being, especially other Mages, that they come across. The most common reason is that Banishers see the entire cosmology of the World of Darkness as they Awaken, including the Abyss, and how it all fits together. After that, they decide that they will use magic, but only to ultimately stop it from ever being used again.
Also a good description of the Guardians of the Veil. Probably the only Well-Intentioned Extremist group that fully acknowledges that status. It's right there in their creed: "Sins for a just end bring wisdom to the Awakened." This self-awareness is probably why they're one of the protagonist factions— they take no pride in what they have to do, only the ends.
WoD Hunters in both Vigil and Reckoning. The average Hunter group has a 75% chance of trying to kill at least one benevolent or neutral supernatural in the name of keeping humanity safe from things that go bump in the night. A fair number of plot hooks in assorted Hunter sourcebooks revolve around overzealous Hunters being tricked into going after the good-guy faction of a given splat by one of the bad-guy factions. They aren't helped by not having access to information about the various supernaturals that could help them sort good from bad.
This game, being set in the World of Darkness, has plenty of non-Lemurian Hoffnungs (those who became Geniuses through a wish to change the world)) who end up becoming this, particularly if their Karma Meter falls too low.
The Tau of regularly employ mass murder and orbital bombardment when a species denies multiple "offers" to join them. The "well intentioned" part is what sets them apart from everyone else - factions such as the Imperium will blow up their own planets and murder billions in the name of mere survival.
Also in the setting are radical Inquisitors who decide that the best way to defeat the forces of Chaos is to turn Chaos against itself. If they're lucky, their colleagues will execute them for heresy; if they're unlucky, they are lost to The Dark Side. If Inquisitors are lost to The Dark Side, many of them still try to serve the Imperium, no matter how many innocent lives are lost in their schemes.
The Eldar are so devoted to the continuing survival of their species that they are willing to sacrifice everybody else to keep going. One of the sidebar quotes in the Eldar Codex even says, "We would rather ten million humans die than one of our own."
Most of the Chaos Space Marines start out this way before the Istvaan Drop Site Massacres. Some still are. Special mention goes to the Alpha Legion, who only sided with Horus because they foresaw that it would ultimately lead to the best possible outcome for the galaxy if Horus actually won. Indeed, they are still loyal to the Emperor, although in the darkest way possible.
Horus himself, at least initially and/or from his point of view. He was presented with a vision of the future of the Imperium as a bloated, oppressive, theocratic state that had forgotten him, discarded the Primarchs, and deified the Emperor. He was told that the Emperor was destroying the benign denizens of the Warp in his quest for apotheosis and would abandon the galaxy to achieve it. This, combined with his bitterness toward his distant father for subordinating him and his brothers to the Council of Terra, made him conclude the Emperor was a corrupt dictator he must overthrow. By the time of the Battle of Terra, he had completely fallen to The Dark Side.
The Lizard Men faction of Warhammer's goals are sympathetic (purging Chaos/Skaven, trying to follow the plans of the Old Ones), but their methods are ruthless.
The Ashbound druid sect believe in the Power of Nature. Well, they are druids after all. The "extremist" part kicks in when you consider that they believe that the best way to protect nature is to ban all arcane magic, burn down all the cities, and go back to living in mud huts. They can be any alignment (well, any of the Neutral variants at least, being mainly Druids and all), so they feature both the genuinely well-intentioned Neutral Good ecoterrorists and the rather less pleasant Neutral Evil ones who just like watching cities burn.
Another of the five druidic sects, the Children of Winter, are dedicated to wiping out civilisation as part of their belief in Social Darwinism. They believe that people can only survive if the weak are winnowed out and the Mourning represents the approaching "Winter" of the world itself. As a result, they unleash famines, plagues, pestilence and all other sorts of natural disasters in the name of "the greater good".
Queen Aurala also fits this. Well Intentioned? Really devoted to her people and wants to help the whole world. Extremist? She will consider every possibility to make her dream possible, including war.
The prestige class "Walker of the Waste" from the Dungeons & DragonsSandstorm sourcebook can be this. They seek to dry out the world. but only to make it a safe place for the people of the wastes, who are so adversely effected by water that some can drown in the rain.
The Guiding Hand is a faction of Well Intentioned Extremists who want to drive all foreign influence out of China, their most iconic fighter being Wong Fei Hong. This may seem like a good thing, particularly because of the opium trade and general imperialism that the Western powers engaged in during the 1800s and the suffering it caused among the Chinese people. On the other hand, Quan Lo, the master of the Hand, is very rigid in his ways, and those who do not embrace them are seen as tools to be used or enemies to be destroyed, making him just as much of a tyrant as Gao Zhang from 69 or Bonengel from 2056. Many Hand adherents despise anything Western or modern, particularly guns and other modern weapons. And Quan Lo's prideful crusade may have some bad consequences — critical shifts shown in the Guiding Hand sourcebook Blood of the Valiant include the extremely stagnant Harmonious Kingdom, where everyone is expected to abide by the laws of the land, technology is outlawed, and All Crimes Are Equal (even rude behavior is punished harshly); and the "Crouching Tiger, Mad Max" setting, which is a post-apocalyptic setting ruled by powerful Chi Warlords who rule by the strongest kung fu.
In addition, there's the Jammers, who have seen the effects of the Buro's control of chi upon much of the populace of 2056, and have decided that no one deserves to be enslaved like this. To this end, they want to destroy all chi so that humanity can be truly free from its influence. There are just a few problems with this plan. One: the primary sources of chi are the Places Of Power known as Feng Shui sites, which often take the form of places like schools, hospitals, and other places important to a community or where innocent people tend to gather, and these guys don't care one whit if these innocent people are hurt or killed in the process of "liberation". Two: the Jammers haven't given much thought to what will happen to the world once all its Feng Shui sites are blown to smithereens, and considering that chi is reportedly tied to life itself, the consequences of wiping out the world's chi may be quite bad indeed.
Johann Bonengel, the president of the aforementioned Buro, also falls under this trope. He entered politics with the dream of creating a world free of "war, ethnic crimes, and personal violence." He achieved these things by creating a worldwide police state whose citizens have few rights and all dissent is ruthlessly suppressed.
The Bronze Faction have managed to royally screw up the Creation in the process of attempting to save it. And yet, their leader, Chejop Kejak, is still convinced that he's doing the right thing, and now, with everything going straight to hell (possibly literally, if the Yozis get their way), he firmly believes the world is better off without the Solars. He is one of the last who personally remembers just how bad the late First Age actually was, but he should also realize that just Terrestrials and Sidereals cannot hold off the Yozis, Deathlords, and Raksha simultaneously, especially with the Scarlet Empress missing and the Realm Defense Grid offline.
Most Sidereals, and many other types of Exalt, tend to end up in this trope eventually. (Then, if the Great Curse addles them badly enough, the "well-intentioned" part falls right off.)
This and Knight Templar are White's schtick. From Ravnica alone, we have the white/green Selesnya Conclave, who wants to keep society together at the cost of individuality; the white/black Orzhov Syndicate, who is much the same without the dryads; and the Boros Legion, whose tactic for building a better world is heavy on the "break heads" side of the scale.
Yawgmoth's archrival Urza would count as an Anti-Hero version. He's still a protagonist, and could even be called the Big Good of his arc, but in his efforts to stop Yawgmoth from taking over Dominaria, he still did such things as start wars, run eugenic programs that produce disposable slaves to be used as soldiers, kill his allies and using their souls to power bombs, and, in one instance, set off an explosion that destroyed a substantial landmass and led to a practical nuclear winter just to kill his corrupted brother. If Yawgmoth's victory didn't spell the horrific torture and subjugation of all Dominarians, Urza would be a straight-up villain.
In Monsterpocalypse, the Radical factions are said to be this. The Empire of the Apes wants to bring harmony back into nature by leveling human cities, but they also protect primitive villages. Subverted with the Terrasaurs: their human supporters believe the Terrasaurs are Gaia's Vengeance, they just find human factories and industrial wastes tasty and are really just looking for a snack.
Starfleet Battles has the Interstellar Concordium, which decided to end the biggest war in the history of the known universe...by occupying the border regions of all warring parties (i.e., everyone else in the known universe). This actually did cause war to substantially abate for a time (while everyone rearmed) which turned out to be a really good thing when the Scary Dogmatic Aliens arrived to wipe out everyone.
Firewall in Eclipse Phase. 10 years ago theFall left earth a wasteland inhabited by killer robots, nanoswarms, and borderline Eldritch Abominations, the total population of the off-world colonies that survived are less than a billion. Firewall is an inter-faction conspiracy dedicated to preventing the extinction of what's left of humanity at all costs, to the point of nuking entire habitats in order to contain X-risks. Their Sentinels (including most player characters) are often sent on suicide missions, multiple times.
The Circle of Orboros in Iron Kingdoms are very similar to the Ashbound and Children of Winter mentioned above; they are a druidic sect who believe that civilisation is an inherently unnatural phenomena that must be crushed and harness natural disasters of all kinds to try and destroy cities and force humans back to nature. They do this because they believe that not only does civilisation interfere with the fundamental powers of nature, if this interference grows too great, the DevourerWurm will return to the mortal world and destroy all humans in order to restore the flow of his powers.
Goreshade the Bastard really, really wants to save his people. In order to do this, he does things like experiment on soulless and sometimes ensouled Iosan children, starts a civil war, transforms himself into an eldritch, signs up with Cryx, delves really deep into forbidden lore and attempts deicide.
The Planescape campaign mentions a rather odd group like this called the Opposers. Their goal is to oppose every political, moral, ethical, and philosophical idea except their own, and they believe this will actually help people. Why? Well, they think that an idea can only succeed and its supporters can only become strong if they have to fight to support it. In other words, opposition makes you strong. As you might expect, this group doesn't have many friends or allies (nor do they want any - they want people to oppose them too). Player Characters can actually join this group if they want to; the only requirement (other than holding to its controversial philosophy of opposing everything) is actually one that makes sense: you must be True Neutral in alignment. (Being Good, Evil, Lawful, or Chaotic obviously defeats the purpose.)
Most members of the anarchic Revolutionary League are viewed this way, at least by non-members. (And this is considered one of the mainstream Factions in the setting.)
In fact, a lot of people would say that all fifteen of the mainstream factions (or at least the most influential members of them) are like this to some degree. (The unnamed editor of The Factols Manifesto certainly claims this.) Their goals are all different, and most conflict with each other, but many members take their philosophies to extremes, sometimes dangerous extremes.