You can do that but it proves nothing because TLS session keys are symmetric, so the archiver can forge server responses and falsely attest that the server sent them.
Look up "TLS non repudiation"
A real solution like TLSNotary involves a neutral, reputable third party that can't see the cleartext attesting to the cyphertext using a ZK proof.
The neutral third party doing attestation can't see the content so they can't easily tamper with it, and attempts to tamper indiscriminately would be easily detected and ding their reputation.
Look up "TLS non repudiation"
A real solution like TLSNotary involves a neutral, reputable third party that can't see the cleartext attesting to the cyphertext using a ZK proof.
The neutral third party doing attestation can't see the content so they can't easily tamper with it, and attempts to tamper indiscriminately would be easily detected and ding their reputation.