How to still cache across sites

I believe that's only a subset of what they're trying to defend against. E.g. the cache isolation feature was discussed in https://bugzilla.mozilla.org/show_bug.cgi?id=1536058 because of an attack that depends only on cached vs uncached load timing. One developer says cache isolation "is a long-term anti-tracking goal too (in order to prevent ETag-based tracking vectors) so this gives us yet another privacy related reason for doing so", so they had already been thinking about this more general solution. There's also stuff like https://github.com/w3c/server-timing/issues/67 where a cached HTTP header can be used as an identifier (which wouldn't be protected by your scheme if you're hashing just the response's body; and you probably can't hash headers without breaking HTTP proxies).

The specific attack mentioned in Mozilla's blog post could be prevented in other cheaper ways, but that would do nothing against a lot of other published and yet-to-be-discovered attacks, so they went with cache isolation to prevent all those different methods (including the timing ones) at once.