If you can't admit to yourself that you've done better than others - or if you're ashamed of wanting to do better than others - then the median will forever be your concrete wall, the place where you stop moving forward. And what about people who are below average? Do you dare say you intend to do better than them? How prideful of you!
Maybe it's not healthy to pride yourself on doing better than someone else. Personally I've found it to be a useful motivator, despite my principles, and I'll take all the useful motivation I can get. Maybe that kind of competition is a zero-sum game, but then so is Go; it doesn't mean we should abolish that human activity, if people find it fun and it leads somewhere interesting.
But in any case, surely it isn't healthy to be ashamed of doing better.
Egalitarianism is the idea that everyone should be considered equal. Equal in merit, equal in opportunity, equal in morality, and equal in achievement.