Mobile OS pushes on both iOS (APNS) and Android (FCM) work the same: push can only be initiated by the developers of an application. So, no third-party server can push an app without intermediation by the application developers. Scheme is this:
Service -> Push service -> APNS / FCM -> Applications
With XMPP, servers act as 'service' from the scheme above, then relay a push initiation to 'Push service', maintained by a client app developers, Push service relays information to a respective service by Apple / Google, who deliver push to an Application, which connects to Service (or does not, pushes can contain all necessary information for displaying in notification, but that's a story for another time).
With email, I'm not really familiar with how IMAP Push works, but the scheme must be the same: an email server must relay info to a push service maintained by app developers, in your case, K9 guys.
From what I was able to gather after reading wikipedia for 5 min, an IMAP server upon receiving an email sends info to some 'mail user agent'. From wikipedia it's not really clear, but it appears that is is supposed to work on end devices. In this case, it won't really work as 'push'.
Google (or any other email provider, like Protonmail, who have their own app) get around this because they control both email server and push server, so it's trivial for them to initiate a push when an email is received.
