In that case, you just unexpectedly told a third party about your request, whereas with HTTPS, you only tell the intended party about your request. How does it improve your anonymity to unexpectedly have third parties intercept your request, read it and respond to it?
If you want to choose to use a third party mirror, then you can do so: just explicitly request the mirror, over HTTPS to avoid a MITM attack.
That's message confidentiality, not privacy. The win to privacy is that instead of telling the intended party and every third party in the middle about you accessing some resource, you only tell the subset of third parties between you and the cache.
Since it's more difficult to intercept access to all possible caches than to a centralised server, that's a win for privacy. At a cost of message confidentiality, of course, but if your message content doesn't need to be confidential (i.e. you're just GETing a resource), it's not a big loss.
> you only tell the subset of third parties between you and the cache.
That's not a privacy win at all if you want privacy from the cache, if you don't trust the cache, if the cache is wholly owned or used by you alone.
Even if all of that doesn't apply, you don't even have a guarantee that it will hit the cache. If it is a cache miss, no “privacy win”.
How many asterisks does this claim of a privacy win need before it should no longer be considered valid?
> At a cost of message confidentiality, of course, but if your message content doesn't need to be confidential (i.e. you're just GETing a resource), it's not a big loss.
You are forgetting the loss of message integrity, as well. That is a big loss.
If you want to choose to use a third party mirror, then you can do so: just explicitly request the mirror, over HTTPS to avoid a MITM attack.