I have to say that achievements really changed my outlook on games. It has made me play things that I wouldn't normally play, and stick with games (relatively) longer. It has also made me, on several occasions, forget about games' stories and gameplay, some of the main reasons you should be playing games.
Could I go without achievements/trophies? Absolutely. On some level (pun not intended), I would thank myself for returning to the reasons why I started heavily playing games in the first place- to have fun and enjoy the story/gameplay. People like stallion83 that have made a career out of accruing Gamerscore is a perfect example of how the process of obtaining achievements/trophies have overshadowed the games themselves.
At the same time, reward systems also inject new life into your games once those stories are finished. When you pick up that $60 game you had on preorder for months, only to play through it's 6 hour campaign in one night, you better believe you'd want more reasons to keep playing it. This is where reward systems shine the most- revisiting your titles to complete that one objective that you wouldn't have thought to attempt on an initial playthrough.
It's a delicate and divisive subject, these non-tangible representations of accomplishment. My struggle has always been to find a balance between driving to whore out achievements/trophies, and playing games just to play and enjoy them. I feel as though I'm at a happy medium now, although recently I feel as though my GS could use a little fluffing...