[etaty]

If I remember, SSH server don't share the public key it accepts. Your SSH agent will try all your key one by one (you can change this behaviour with custom config)

[problems]

Yes, but similarly if you have a given public key you can ask a server if it accepts that key with no need for the private key.

Basically, don't reuse keys in places where you might not want to be identified and use ssh configs to prevent announcing all keys to the world.