I play a fair few games on the 360 and while there may be some achievement whores out there, I don't think they represent the majority. I see achievements as another collectible for if you want to 100% complete a game. I'll only specifically seek out an achievement if I like the game enough to want a reason to keep playing it.
Achievements can really expand the longevity of a game and reward you for certain things that might otherwise have gone unnoticed. Like if in an FPS you manage to kill 10 people with just a sock. Without achievements, that event would go unnoticed.