This can't get better (than having trust in a third party) as being able to prove these things actually goes against one of the common security guarantees of th

This can't get better (than having trust in a third party) as being able to prove these things actually goes against one of the common security guarantees of the cryptography protocols used by TLS. You need the server to actually play along and give up their right to deniability, such as by supporting the new Signed Exchanges protocol being worked on for AMP.