Several controlers set quite intricate Cache-Control headers in order to
hopefully not be cached by any intermediate proxies or local caches. Unfortunately,
these headers are processed by ActionDispatch::HTTP::Cache in a way that squashes
and discards any values set alongside no-store other than private:
https://github.com/rails/rails/blob/8015c2c2cf5c8718449677570f372ceb01318a32/actionpack/lib/action_dispatch/http/cache.rb#L207-L209

We want to preserve no-store on these responses, but we might as well remove
parts that are going to be dropped anyway. As many of the endpoints in these
controllers are private to a particular user, we should also add "private",
which will be preserved alongside no-store.