The cryptography provides nothing to establish that this separation is actually being maintained and there is plenty of evidence (e.g. Snowden) of governments doing exactly the opposite while publicly claiming the contrary.

On top of that, it's a timing attack, so all you need is the logs from both of them. Government gets breached and the logs published, all the sites learn who you are. Government becomes corrupt/authoritarian, seizes logs from sites openly or in secret (and can use the ones from e.g. Cloudflare without the site itself even knowing about it), retroactively identifies people.