EDIT: See comment below this - it appears I might be talking out my bottom.
It's easy to change an app's background ability in Android. By default apps are forced into the background, but if you change that your k9 email will notify you all day long (it does support IDLE)
There's an issue on their tracker that discusses implementing the notification, and docs that say that without it K9 can only check mail every 15 minutes or so, even if excluded from battery optimization. So I don't know what support for IDLE and ‘background ability’ you're talking about. Before Android 9 or so, sure—it could just run in the background and connect to the servers whenever it wanted.
It's easy to change an app's background ability in Android. By default apps are forced into the background, but if you change that your k9 email will notify you all day long (it does support IDLE)