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).
Mail polling doesn't require accuracy. The OS won't delay your polling timer more than 10 minutes in the most aggressive cases, which is perfectly fine.
I'm sure the third party you share your email just to get push notifications with isn't monitoring your mailbox to the second either...
They probably are. If they connect via IMAP, they can use the IDLE and/or the NOTIFICATION command to get notifications of new messages arriving. This is generally as prompt as it gets; the server doesn’t wait much before sending the notifications.
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).