Hard one.
My top three RPGs that got me back into things after burning out on Final Fantasy clones was Resonance of Fate, Eternal Sonata and MagnaCarta 2. The first is getting a nice remake as we speak, and for the second then the PS3 version is probably the superior one here. To that end I don't know if I would suggest the 360 as the path of choice here, not to mention a lot of what I would mention also appeared on the PC with nice mods and better lots of things.
For the 360 it was more that it seemed like every month I was getting new and interesting stuff to play, maybe not stuff for the ages* but still stuff, and local co-op and LAN play was still a thing -- if nothing else Resident Evil 5 got played all the time because of that. Skate 2 and Skate 3... if they and all their DLC crashed into each other and I was left with a combo of both it would undoubtedly be the best skateboarding game ever made but as it stands then vanilla skate 2 or skate 3 with all the DLC is a pretty nice thing to have, something I have not yet been able to find an equivalent for (though it seems there might be something). Similarly despite having a gaming PC for most of the 360 lifetime I never felt like I was missing out on playing the vanilla versions of games on the 360 -- today you would pick a PC to play Skyrim but when it dropped I did fine with it.
*will I particularly need to play Deadly Premonition again? Will I need to thrust it upon friends and say play this? Probably not on both counts but definitely glad I was around to see it. Same for things like Two Worlds 2, Venetica, Alpha Protocol (which I was told when writing that has just been yanked from Steam due to rights issues -- my disc works just fine though), Vanquish, the Risen series, Ninety‑Nine Nights and Kingdoms of Amalur.
Here is the 360 end of my game shelf. I have a few games elsewhere right now (the resident evils are just out of shot with all the other resident evils, as is I think prince of persia), skate 2 is gone, as is forza, some mx vs atv titles (though I like the original xbox for those) and some other things I think I suffered the younger sibling effect for. Some of it I have for being nice examples of a mechanic, first new mechanic,
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So yeah I think it was a combo of easy modding, local co-op, single player still reasonably being a thing, some people doing some nice experimental stuff, microtransactions not being crowbarred absolutely everywhere to the detriment of games themselves (look around say a Halo multiplayer and they will try to stripe you up for everything from more space to store your replays to certain skins but nothing like we see today) and enough of it all happening that to this day I still have a tiny bit of a backlog.