While San Andreas and GTA 4 share some similarities about a character trying very hard to change their output in life, I'd argue that 3 and Vice City went through vastly different escalations.
Tommy Vercetti's rise to being a literal kingpin is very different than other 3D GTAs, but one of the reasons I really resonated with GTA 3 over the other 3D GTAs, even though it came first, is because Claude's story matches the concept of GTA better than any other protagonist.
Claude is a Mr. Fix-it. A Hitman for hire. And he gets the job done no matter how or when. He was never in it to rule the city (like Tommy from Vice City) or somehow ambitiously reach a dream of another life (like C.J. and Niko). Claude was simply good at his job and got paid, no questions asked, as long as someone was needing a favor and could pay what was owed.
Yes, there was simplicty to this story, but it worked. It used the characters around him to progress the plot and your actions only added gasoline to a fire that was already lit. However, that's not Claude's problem, it's theirs. The moment it became his problem, he fixed it.
He endures a slight moment of betrayal in the end that ends in gunfire. Then he realizes the person he 'saved' in the end is worth too much trouble, so as they're both leaving the screen cuts to black mid conversation with a gunshot sound, solidifying for not only him, but also the person playing the game, that for Claude the job came first...never get too close, never get involved, and don't fully trust anyone.
He is a killer, a mr. fix-it, a man for hire, but again, this completely lines up with the gameplay loop of Grand Theft Auto.