STAGE SELECT:
I was thinking you wanted the "best" console launch games, and that theme would have led me to a lot of of the games mentioned before my comment like BotW, Halo, Super Mario 64 etc cause you can't deny how much those games meant for their respective systems. But my personal favorite launch games? Here's that list:
Spider-Man 2 (DS): Look, it's not a great game, but for whatever reason I played the shit out of it and always looked forward to side swinging through its generic level design as sort of a brainless escape. It's one of those games that does absolutely nothing new or innovative, yet is so easy to pick up and play I would do so over and over again. Sometimes you just want run-of-the-mill action. Or at least I do. It definitely held me over in those early DS days when it was either this or playing Super Mario 64 DS with a tiny D-pad.
Peter Jackson's King Kong (360): Ok, this is cheating because I never actually played the 360 version, but I did play the PS2 game I assume this is a remaster of and let me tell ya. This is a fucking good game. Hell yeah I want a cinematic experience in which I play a giant gorilla that rips dinosaur mouths in half. Don't let this game be forgotten. It is a treasure. There's even a Polygon article completely overhyping it: https://www.polygon.com/2019/7/8/20680953/king-kong-xbox-360-game-rayman-michel-ancel
Wave Race: Blue Storm (Gamecube): I mean, I could put Luigi's Mansion here since I actually own it and do enjoy it, but it's always been a bit of a slow game. But when the Gamecube first came out, this game's water graphics and physics blew my 11-year-old mind.
I've heard people (specifically, YouTube channel Game Sack) say the controls are more stiff than Wave Race 64 and they don't like the sarcastic announcer in Blue Storm and I definitely believe those criticisms despite never playing the N64 game. It could be a bit hard to make some of those turns. But damn I loved renting Blue Storm. So fun and graphically hypnotizing.
CAGE MATCH:
I have played far too little of KOTOR to have an informed opinion, but I've never played anything of Tales of Vesperia. I'm gonna vote for Tales of Vesperia cause that game is not attached to a fan base that annoys the living fuck out of me.*
*: Star Wars fans in general; maybe the KOTOR-specific fan base is fine, but my God has the more general SW peanut gallery soured me on that franchise.