People may Google my name and find things I don't want them to see.[1] Fortunately, I can do the same to them. Even though most people are nice most of the time, this fact incentivizes everyone to be nicer than they otherwise would be.
If we allowed people to take down results they deemed undesirable, it would give bad people fewer reasons to behave nicely. After all, they can just make search engines remove the entries they dislike.
And there's the problem inherent in all censorship: Who do you trust to determine what you're allowed to read or say?
1. Actually they might think I've been murdered, but that was a different Geoff Greer (may he rest in peace). Should I be able to get those results removed? After all, the late Geoff Greer won't mind.