The idea behind private keys is that they are private and never sent anywhere so I believe your assertion that the server knows anything about your private key is incorrect.
Do you know where the "don't put hot food in the fridge" idea came from? My thought has always been that if my fridge warms up the compressor will fire up and cool it down again.
Here's a link to Yubico with a visual diagram of how passkeys work: https://developers.yubico.com/Passkeys/How_passkeys_work.htm...
But fundamentally it's very similar to how all public/private stuff works. You send people the public key and sign stuff with the private key.