Coming back to this 7 days later I think I understand what you mean.
The site content is authenticated with the symmetric key.
The symmetric key is authenticed with the private key.
That led me to jump to the conclusion that the content is transitively authenticated with the private key. But the security model of symmetric and asymmetric authentication schemes is of course very different. The server's private key is never disclosed so you cannot use it to forge the key exchange. But the symmetric key is of course shared and only defines a single level of trust: You either have it, meaning you can both verify messages but also sign arbitrary messages. For the purpose of transport security that is perfectly fine as both parties have full trust in each other for this purpose (the whole point is to guard the connection against anyone else), but for public authentication you need an asymmetric structure that allows verifying without the ability to sign. Symmetric cryptography by design does not enable this.
It is irrelevant that the symmetric key is authenticated with the server's private key because the presupposition that the symmetric authentication works is flawed. Verification by the untrusted public is a different scenario from verification by the two trusted parties of the communication and the presuppositions do not carry over.