I know it's a controversial thing, but I don't believe Goku was as instrumental in Vegeta’s development as people say.
As far as we know, the two only met a dozen times before the Buu Saga. Meanwhile, he lived with Bulma for years and had a child with her whom he later at least trained (meaning he spent time with him). We also see Gohan showing up at the CC and Vegeta acting like it's a common occurrence.
And Vegeta is almost as distraught about Gohan's death in Buu Saga as he is about his wife and son.
Goku may have been a push in the right direction for Vegeta, but I'm certain that Bulma, Trunks and Gohan were much more important in his redemption.
I don't usually talk about Goku but I feel like my little silly hours post has fallen into Media Analysis Territory, and I love me a Media Analysis, so I'm gonna talk about Goku (based on how Toriyama wrote him) under the cut, but the TL;DR is:
Goku (the way Toriyama has always written him) is just here to have a good time getting good, and the credit for his very potent (and distinctly passive) influence over people is squarely, "This Man Showed Up and Was Strange In A Way I Didn't Know People Were Allowed To Be" and That's It. That's Goku's Influence. Being a Freak on Main. Because sometimes that's all someone needs to start questioning their whole worldview.
Toei's Goku is "This beautiful man reshaped my life with his bare hands and I think about him every day"
Toriyama's Goku is "I was giving this guy a ride home and while we were waiting in a taco bell drive thru he said something so whimsical and unhinged that after I dropped him off I stayed up all night thinking about it and ended up watching the sunrise for the first time since I was like six years old and now I might take a roadtrip and get a tattoo"
anyway, MUCH longer analysis under the cut for interested parties (again, citing the manga specifically) because I love talking about storytelling
1) Yep! Piccolo credits Gohan for his change of heart
2) Yep! Krillin is credited for the androids redemption, by most of the cast and both 17 & 18 themselves
3) Meeting Goku is generally written as an Event that changes the course of someone's life because he is Distinctly Strange as a person. Vegeta said that Goku's influence made him Softer, and Piccolo said that Vegeta's growth is a result of having met Goku, which changed him. Both are Factually True Statements, as Vegeta would still be out and about doing space gangster shit if he'd not met Goku.
4) To that point, Goku's influence on people is extremely passive, not active. It's a running gag throughout all three titles that Goku is Largely Unaware that his non-combat actions affect other people (until he is told about it, is either confused or earnestly apologizes, and then does it again). He is constantly doing whatever he wants with little, if any, consideration for the consequences, and it's something he gets called out on in all three titles.
5) Goku's influence on Vegeta specifically is just existing as a living testament to how everything Vegeta's ever known is a Lie, including but not limited to the Super Saiyan legend (Goku's power and personality being so out of line with what Vegeta's been told about the Legendary Saiyan is what makes him start questioning the truth of it).
More importantly, all of the things Vegeta grew up understanding as weaknesses, Goku casually and consistently proves are strengths (love, trust, empathy, etc), which Gohan is not as influential about early on, because (as Vegeta points out) Gohan cannot access that strength yet except when he is consumed with fury, while Goku can do it with his Soft nature intact. Observing someone else be open hearted, while still managing to be secure and strong, often gives the observer permission to open up to that experiences as well, which is what eventually happened to Vegeta (and that observation is not only of Goku, of course, but seeing another Saiyan doing it was important when it came to Vegeta, who regularly used his heritage and rank as an excuse to not take the softer road. Vegeta undermines Goku's lineage with his soft Earth upbringing, but it doesn't change how strong he (or Gohan, or Trunks) is as a Saiyan, which puts Vegeta in a corner and forces him to either bite the bullet and start giving a shit on main or admit that it's not a Saiyan Thing or a Power Thing and he's just afraid to be attached to the people he's kicking and screaming about definitely not being attached to).
That's what the influence is, and why it matters. It's not Goku personally scratching his enemy's chin and giving them a cookie every time they opt out of murder. It's the simple act of being Happily and Unwaveringly Himself in plain view of people who didn't know that way of life was a viable option. And now that they do, they may or may not be willing to risk trying out for themselves, given the chance (which Vegeta did when Bulma offered him that choice, and ended up liking it so much that it freaked him out, because Goku is only an influence, not an instruction manual for how to live like that. That's the joke about him being the tool itself, and not the muscle or the mind to make it mean anything).
6) Goku doesn't just spare his enemies. It's worse than that. With rare exceptions, he actively works to avoid killing enemies, with the verbal purpose of wanting them to get stronger, so they can fight again, because he enjoys it. That always throws a Big Bad off, because it is an absolutely insane thing to say to someone who wants to kill you. Goku is usually upset when he has to kill someone -- but not because murder is Wrong, or that he's morally opposed at all -- it's because he thinks it's a waste of a fun rival.
The difference between sparing someone who hurt you out of moral superiority and sparing someone because you had a good time almost dying just now is very important to how unhinged Goku looks to his enemies (and his allies).
Additionally: Sparing someone's life is a pretty important factor in how those characters continue to live when they expected not to, which is a very influential life event, for better or worse.
7) Most of the credit Goku gets for changing the ways of his enemies (in the manga) are from Dragon Ball, in which it was a very young Goku's spirit and determination that inspired his antagonists to live more like him (because Goku is and always has been very simple in his worldview, and doesn't hold a grudge. If you're a bad guy, stop doing that. You should try eating something instead. If you stop killing people we can fight more and it'll be more fun if neither of us dies. And he's still like that, and it's still confusing to everyone around him, because DB is a comedy at heart).
Krillin convinces Bulma in the Android saga to not destroying the androids early, because it gives them a common goal with Vegeta, since having a common goal is how everyone there became friends. He credits having to work together as the thing that changes enemies to allies (which was his experience with Vegeta on Namek, and his experience with Piccolo against the Saiyans), and just acknowledges that all of them met through Goku, which they did.
Roshi also does this in Dragon Ball, crediting Kami for the entire adventure and the fact that everyone became friends, because without the Dragon Balls, Goku and Bulma would have just stayed home that day and none of it would've ever happened.
8) Goku actively rejects credit in Super's manga (probably because the anime and the movies kept insisting he is superman doing Good and Just and Right in the world), as, again, he doesn't do any of it with the intention of Doing Good. And, again, the running gag throughout all three titles is Goku being Blissfully Unaware that his choices affect other people.
9) If Goku gets undue credit, it's largely from Toei and from fans of the anime/older movies, which constantly revised dialogue and added filler to give Goku credit for things Toriyama wrote him having nothing to do with.
Because the shows follow the manga's plot while still making it their mission to frame Goku as a Leading Man Action Hero, it adds up to a lot of weird plot gibberish, excessive padding, and nonsensical storytelling in order to frame Goku as Irrefutably Good, where he absolutely lacks any rigid sense of justice and morality in the manga. He's just out here having fun, and he gets mad when you play the game wrong (and hurt people he likes).
tl;dr part 2: Goku's influence over villains can be summarized by the vine of the guy in a mcdonalds pointing at a couple and going 'is this allowed??' but instead of a couple it's just Goku genuinely and truly having a wonderful time while being in extreme danger and it's mind-bending that someone like that can not only exist but also thrive