So here's something I can't get a straight answer on:
Surely if content is served over HTTPS with a valid certificate, it should be possible to save (possibly as part of a WARC) a "signature" of the TCP stream that would go beyond proving that a web archive was created at a certain time, but also that it was served using that person's private key and thus from that person's web server. To claim otherwise, the subject would have to claim that a fraudulent certificate was generated for their domain or that their web server was broken into.
Basically, the way the crypto math works in HTTPS is it's a symmetrical proof that only proves that either the sender or the receiver sent the TCP stream. Normally that's OK, because you trust yourself. But in this case the problem you're trying to solve is to prove what happened to a third party who doesn't trust the receiver, so your idea doesn't work.
It's the same with the RSA key exchange. It's inherent in the fact that the TLS negotiation exists to make both sides agree on a common master secret (and some public cryptographic parameters like which cipher to use), from which all the keys used to encrypt and authenticate either direction of the stream are derived. Once the master secret is known, all keys are known and the rest of the connection can be decrypted and/or forged at will. (The "triple handshake" attack exploits this, by making two connections share the same master secret.)
The certificate is used to sign (parts of) the values used to create the master secret. It doesn't sign anything after that.
Surely if content is served over HTTPS with a valid certificate, it should be possible to save (possibly as part of a WARC) a "signature" of the TCP stream that would go beyond proving that a web archive was created at a certain time, but also that it was served using that person's private key and thus from that person's web server. To claim otherwise, the subject would have to claim that a fraudulent certificate was generated for their domain or that their web server was broken into.