There's no magic bullet. As soon as your template references user specific (private) data, the caching advantage almost disappears, because each user requires their own cached rendering. You might as well cache it on the client and save the round-trip, and any additional network/server performance issues.
Rendering public, read-only content such as a tweet, however, makes total sense to cache on the server.
JSON is still rendered. It's not "render or send JSON", it's "render a string that contains formatted display output A, or a string that contains structured data B".
Facebook probably requires a lot of per-user effort true. A lot of sites (github for example) probably don't. It's probably limited to a couple areas of the header generally, or an A or B conditional render for an ownership page.
Rendering public, read-only content such as a tweet, however, makes total sense to cache on the server.