> If I had a mail client installed, couldn't the app just periodically connect to my mail server, see new mail was available, and then pop up a notification so I'd know?
Desktop mail clients do. Phone mail clients can’t, so you either check on a server, or don’t get notifications on time.
Mobile platforms seem a bit broken. Timer/alarm apps seem to be able to take actions and notify on a regular schedule, it is specifically scheduled network activity that's restricted?
yes mobile OSes will delay and group background activity especially if it involves network access. But we're talking minutes, not hours of delay... Some phones are very aggressive and the app will need to show a persistent notification to keep background polling working, though.
I've never used a mobile client that used a third party to monitor a mailbox, they all do polling as you suggest. I'm not sure the man in the middle approach it's as common as OP is implying here (for non-first party clients).
Desktop mail clients do. Phone mail clients can’t, so you either check on a server, or don’t get notifications on time.