Lol, it's not like Rider wasn't allowed to hide his identity. He just didn't want to.

As for your other cases, i'm pretty sure the servants themselves prepared arrangements to hide their identities, it wasn't anything that was given to them.

Saber's NP is a trade off, like it was mentioned in fsn before the more famous a spirit and his NP is the more powerful it is but it is also more easily identifiable.