The notifications are flakey from the Mail App, when you set it as running in the background you do get a variable delay with the Mail App closed, seems to be true of Windows 10 and 11, currently.
Latest info Win 10 on 3 accounts is around 1 min to 30-60 min delay to get the notification with Mail App closed. Same on Win 11 but is around 5 mins.
This is rather a small real world sample.
With 6 Accounts setup over many years, variations in notifications and the badge marker of unread mails on the Taskbar Mail icon.
Does not bother me as I open the Mail App frequently, notification variations are a very minor point.
There is no instance in the System tray bottom right, the Notification Icon very far right fills in on receiving mail and if you click on it you get a preview popup as normal. Also badge marker on the Taskbar icon for the Mail App, also normal.