IMAP works amazingly well if you're using dovecot as a server (including search and push e-mail), and CalDAV/CardDAV also work well if your server actually implements things properly, and projects like Horde have shown you can run and deploy web-based services for them at scale (even though I still regret they're using PHP).
Sure, anything related to DAV has more than a few warts, but my main point here is that things like JMAP won't cater to an installed base or deliver benefits across the board until they finish filing all the corners on their shiny new... cube? Dodecahedron?