1) Most websites are actually web applications which have to maintain state via cookies over a TLS secured TCP connection. This means that any indirection must involve tunneling TCP connections, which timeout fairly quickly and require several round-trips to setup.
2) Even when websites are read-only and have no state to maintain, there is no common way for a peer to prove that content is authentic. Unfortunately, the authenticity 'guarantee' that TLS provides does not transfer to a third party [1]. There is a proof of concept server implementation for non-repudation over TLS [2], but every website would have to choose to implement it, and there aren't many incentives to do so.
The best you could do here is trying to reach a consensus about what the content of a page is, but you would have to account for subtle or not-so-subtle differences between requests by different clients at a different time.
[1] https://crypto.stackexchange.com/questions/5455/does-a-trace...