SMTP/IMAP isn't better than webmail. IMAP is an awful protocol. It's less secure than HTTPS. It doesn't do a better job of avoiding "spying" than webmail does. You can sync multiple clients with IMAP, but webmail accomplishes the same task so seamlessly clients don't even need to be aware of it.

IMAP is a pretty good example of the kind of protocol that should be eaten by HTTP/RPC APIs.

(I've written a couple IMAP implementations, both client and server, and was [in the 1990s] responsible for the mail infrastructure for a popular ISP.)