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Vegeta is behind everything. Everything. Read the page for more details than your mind can handle.
It also could be stated that had Piccolo simply ignored Raditz and simply shut up about the Dragon Balls, a large portion of the conflict would have been avoided in DBZ. I suppose his eventual demise was a form of Karmic Death.
Of course, by the time the androids arrived everyone would have been significantly weaker had the Saiyan and Namek sagas never happened, and everyone probably would have been killed.
The androids themselves might have been weaker too, since they were designed to kill Goku, and Dr. Gero would not have had access to data or DNA from the other Saiyans or Freeza and King Cold. Of course, none of this would have prevented Babidi from coming, and then everyone would have truly been screwed. (He possibly might not have gathered enough energy to resurrect Buu, but he still had Dabura...)
The arcs with Buu and Babidi are only shown to occur in the main timeline where Future Trunks did a lot of unnecessary time traveling (stopping Mecha-Frieza and King Cold being the first). That in itself plays the trope straight, but again, the events of Future Trunks' timeline (and most of DBZ) would not have happened if Piccolo didn't tell Raditz about the Dragon Balls. Nappa, Vegeta, Mecha-Frieza, and King Cold would have never come to Earth for revenge and Goten and Trunks would have never been born. Piccolo would have never merged with Nail or Kami and Dr. Gero wouldn't have the data to make the Androids as strong as they were. And without everybody needing to get so ridiculously strong to survive the previous arcs, Babidi would've presumably had to look to some other planet to get enough energy to take a shot at reviving Buu.
On the other hand, it was made apparent that Frieza's empire wanted to conquer Earth and sell it, so Nappa and Vegeta would likely have been sent to Earth anyway. If not them, then some of Frieza's other soldiers.
Not really. Earth was some backwater little planet where Bardock sent his son to keep him safe from Frieza. Raditz only went looking for Goku because he was his brother, and Vegeta and Nappa came because of the Dragon Balls. Considering the three usually worked as a team, the fact that Vegeta and Nappa didn't show up with Raditz meant it was more of a personal errand for him than anything to do with wanting Earth. Frieza's organization might have gotten to Earth eventually, but it was just one of countless inhabitable planets with nothing special about it other than the Dragon Balls. If nobody off-world knew about those, it would've remained far beneath the notice of Frieza and his top minions. Worst-case scenario would probably be some of those nameless mooks with the wrist-mounted blasters being sent to conquer it, and even Chiaotzu could handle those losers.
Kami making the Dragon Balls in the first place. He made them to make the world better but people only wanted them for selfish purposes (Pilaf taking over the world, Vegeta wanting immortality, Bulma wanting a boyfriend...).
Piccolo has another one when he fights 17 for the second time. Because he powers up, Cell can locate him and therefore the androids, speeding up his plans to absorb them. Had he waited for Vegeta and Trunks, 17 and 18 would have gone the same way as 19 in half the time, before Trunks takes out Cell like he does upon returning to the future.
Even moreso than Kami making the regular Dragon Balls; how about the Black Star Dragon Balls? They work just like the original ones did, only they scatter over the entire galaxy when a wish is made, and rather than turning to stone for a year, they destroy the Earth in one year's time if they're not all gathered and brought back! I know you made these before you expelled your evil side, Kami, but still, what were you thinking?! Or was that just your trial run?
The entire premise of the Shadow Dragon Saga in GT. The threat is their own fault for using the Dragon Balls too much throughout the history of the show. Then again, GT has a serious Canon Discontinuity problem, so this might have to go on that page.
Bulma, for removing 16's internal bomb, though Cell notes that since he's so overwhelmingly strong at this point; the blast would either blasted him into easily regenerative pieces or not affected him at all.
Not to mention, given how fast the Z fighters can move, Goku grasps the Idiot Ball moreso by simply not just teleporting Cell, grabbing King Kai and his pets, and then teleporting the fuck out. It would have prevented their deaths, and even with Cell coming back, they would have had another fighter to even things out.
According to the movies, Goku's Senseless Sacrifice against Cell was an even bigger example, as Cell's explosion freed Bojack from imprisonment.
Buu was even worse, and would never have gotten as far as he did without the heroes. The Supreme Kai allowed Gohan to be drained of his energy, vastly underestimated how much of the necessary energy would be extracted. The battle between Vegeta and Goku produced so much power it accelerated Buu's awakening even more. Piccolo's attempt to buy more time for Goten and Trunks prompted Buu to kill every person on Earth except for the main cast with one attack. Gotenks acts like he's completely outmatched and that he's going to be defeated as a setup for his Super Saiyan 3 transformation, but Piccolo takes him seriously and destroys the door to the Hyperbolic Time Chamber, leaving them trapped forever after Buu escapes anyway. And finally, when Goku and Vegeta successfully rescue their sons and Piccolo from within Super Buu, Vegeta gets the bright idea to also yank out Fat Buu, causing the monster to revert into Kid Buu, who was much weaker than Super Buu but evenmoremonstrous.
How about GT? It turns out that the heroes can only make one wish with the Dragon Balls for only a year. Thanks to overusing it from Dragon Ball to Dragon Ball Z, that's how the Shadow Dragons came into existence.
It gets worse when the Goku and Vegeta's fusion, Gogeta, starts playing around with the final boss Omega Shenron and loses his chance to finish him off. Goku realizes he became extremely cocky as Gogeta and and spends a entire episode trying to fuse again since the duo can't defeat Omega without being combined.
To be fair with that last one, Gogeta was pissing Omega Shenron off on purpose so he'd unleash his Negative Karma Ball in an attempt to kill them and everything else, which Gogeta could then turn it into a Positive Karma Ball and use it to destroy the shroud of minus energy covering the earth. When that job was finished, they went for the kill—only to run out of time.
Especially during the Cell and Buu arcs - quite a few problems would have been solved if the Z Fighters hadn't gone in with both guns blazing.
The Supreme Kai's plan for stopping Babidi from gathering enough energy to resurrect Buu involves... bringing individuals with a lot of energy right to him, as well as straight up allowing him to acquire Gohan's energy without a fight. Though Babidi's plan has a few hitches, it succeeds without sacrificing anything Babidi cares about. Babidi takes great pleasure in rubbing in to his near-dead enemy how Supreme Kai pretty much handed him victory on a silver platter.
Ultimately, with the sheer regularity of both this trope and Goku's displays of mercy to fallen foes, it's a wonder how often things work out to where the Z Fighters keep gaining allies in the endů
In the Buu Arc, when Vegeta and Goku are in Buu's body, Goku deliberately shouts out that he and Vegeta can't fuse together anymore, in front of Buu.