It would also be nice to extend IMAP to support UnifiedPush...

The use-case would be new mail notifications; it wouldn't be too hard to do, and would be quite useful, as I typically have multiple e-mail accounts.

You could however build a UnifiedPush-compliant SMTP-based or IMAP-based distributor.

JMAP has a push implementation I really like, and it's natively compatible with UnifiedPush. Would be nice to see more adoption there.

https://jmap.io/spec-core.html#push

Notifications would work (and does), maybe a sieve rule for deleting old messages?