No, such logs would not prove (to a third party) that the given file was served by someone holding the private key.
TLS works in two phases:
- The server makes use of its private key to prove its identity to the client and negotiate a session key.
- The session key is used to encrypt and authenticate the application data.
The client and server both have access to the session key. The server's private key is not involved in the second phase at all.
So, having negotiated a genuine session key with the server and captured a genuine handshake to prove it, a malicious client could then proceed to use this session key to falsify the rest of the log and claim that the server sent it a different response.