Answer to your edit: team names are not private, as their hashes can be found in our merkle tree, which anyone is free to mirror or lookup. So if you name your team `foobar` someone would be able to hash foobar and look it up, and yes, prove it exists. This is a feature, not a bug :-)

But if you picked a team name with very high entropy, then I guess that would be semi-private in that it would be undiscoverable by brute force.