> I was kinda hoping to avoid storing the private key on the local machine at all.

I may be misunderstanding but I use U2F with OpenSSH (8.3). The private key is not on the local machine. It's still SSH public/private key pair but the private key on the computer is only a "key handle", not the real private key. The private key is protected on the security key. I don't mind copying that private key around: although it looks like a private key, it's just a key handle and not the actual private key.