When someone gives an answer, it may not be the complete solution you need. If you couldn't tell their name, how would you know that the comments you're getting in response are from that user? And how would you tell whether two answers were posted by the same person or not?

Mechanically speaking, it also promotes some strange effects when you consider anonymous users and viewers. To set this up proper, we either have to restrict it to people with accounts, or we enforce it on anonymous users. The former is a bad idea because creating an account is an enabling act, which removing the ability to see usernames is the opposite of enabling. The latter is a bad idea because it makes this community-driven site look very impersonal and robotic.