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More than a few fans have stated they wished that Midoriya had never gained powers and that the series was about him becoming a hero despite his lack of a Quirk. Horikoshi stated that this was indeed the original plan for the manga, but it was scrapped because he felt it would be too unrealistic for him to keep fighting enemies with increasingly outlandish and overpowered Quirks, which All Might explains to him at the beginning of the manga.
For a series that centers around a hero school, it doesn't take advantage of such a setting all that much. This becomes a lot more obvious later on with the confirmation that the series has entered into its final arc. A final arc where Midoriya and company still haven't even completed their first year at U.A.
Relating to the above, the non-combat aspects of heroism take a backseat to fighting villains much of the time. Establishing a unique persona, setting an example for the public, and how to manage your own agency are all things we’re told are important for heroes, but only get explored during OVAs or blink and you’ll miss it scenes between major plot points. Even pros who behave and get treated more like celebrities such as Hawks and Mt. Lady are shown doing field work more than anything else. Given the nature of the show’s genre this is to be expected to a degree, but it’s odd to establish and reiterate these elements without having them play a more central role in the story. This becomes even more obvious when public relations between heroes and the public go sour after the villains' plans begin to take off and there's no mention of hero management trying to repair their reputations, despite that being their job. This actually receives some light Lampshade Hanging in the final arc, where after receiving basically zero focus throughout the series, the students in U.A.'s management classes take a stand to livestream the final battle and show the world that they're still trying to fight, and admit that they really should have done something to support the heroes before now.
Uraraka's confession that she went into heroism to support her family touches on an issue of to what degree heroes should do their work for compensation. While some heroes are motivated by fame and fortune more than helping others, heroism is a demanding and dangerous job, and heroes still need to be able to provide for themselves and their families. Unfortunately, Uraraka gradually becomes motivated to be a hero for heroism's sake, and while this can be considered Character Development for her, it ultimately discards a potentially interesting ethical debate.
Paranormal Liberation War Arc
During the war, Midnight orders Yaoyorozu to create a sedative to help immobilize Gigantomachia, an action she says is illegal but is necessary given the circumstances. This, along with Hawks killing Twice, could have been used as an example of heroes being forced to disregard the law in order to protect people, and it might have been interesting to see the U.A. students' thoughts on this (especially Yaoyorozu, who participated in the Bakugo rescue mission, albeit reluctantly), but the questionable legality of this act is never brought up again.
Ironically, the Liberation Front didn't get much of a spotlight in this war. The entire army was trapped in the mansion fighting the Heroes and they were likely cut to ribbons by Gigantomachia when he made a move. It would've been nice to see the Liberation Army clash with the heroes in the city, adding to the confusion and panic of the battle with Shigaraki. As it stands, it made that entire arc worth of build up about an army of quirk using zealots seem inconsequential.
After the war would have been a potentially good opportunity to show a student dying in the line of duty as a way of underscoring the dangers of hero work. Despite that, while several students are injured and Midnight, a teacher at U.A. is among the dead, all the students make it out of the war alive.
The possibility of U.A. having a traitor was brought up by the teachers, but quickly forgotten about. Between that and the time that Yuga Aoyama was revealed to be the traitor, he hadn't taken any further traitorous actions, and none of the staff nor students were shown doing any investigation. Not helping is the way the traitor was discovered: Tooru (a minor character) just happened to be looking for him and coincidentally found him with his parents begging him to go along with All for One.
Tamaki's hyped-up ultimate all-out attack ends up doing jack squat to the powered-up Shigaraki, leading to Bakugo's Heroic Sacrifice. Sure, he's not a major character and this just shows how grave of a threat Shigaraki is, but a great percentage of readers wished that he could have done at least some damage, considering how awesome and potentially powerful his Quirk is.
The discrimination towards those who had their bodies morphed due to receiving heteromorphic Quirks (an issue that revolves around the League of Villain's Spinner) had a lot of potential for some interesting storytelling. However, said discrimination has just been something that goes on in the background as it was never given a Story Arc to truly delve further into this overall issue. This comes to a head during the Final Battle where Spinner is seen leading the charge for his heteromorphic brothers against the heroes, which left many readers feeling that there was no build-up whatsoever for this plotline to have gotten to this point. Thus, leaving people feeling no investment in it. Adding to this is the two lesser developed heteromorphic Class 1-A students, Koda and Shoji, facing off against Spinner during this battle, which similar to the original issue, people were left feeling that they would be more invested in these characters had Spinner, Koda, and Shoji all received development earlier that delved into the discrimination towards heteromorphic people.
. Even pros who behave and get treated more like celebrities such as Hawks and Mt. Lady are shown doing field work more than anything else. Given the nature of the show’s genre this is to be expected to a degree, but it’s odd to establish and reiterate these elements without having them play a more central role in the story. This becomes even more obvious when public relations between heroes and the public go sour after the villains' plans begin to take off and there's no mention of hero management trying to repair their reputations, despite that being their job. This actually receives some light Lampshade Hanging in the final arc, where after receiving basically zero focus throughout the series, the students in U.A.'s management classes take a stand to livestream the final battle and show the world that they're still trying to fight, and admit that they really should have done something to support the heroes before now.