Not Receiving Push Notifications of Email

I have an iPhone SE running iOS 14.2.

My mail app is not fetching new emails or sending notifications until I actually open the app. Even if I've received 20 emails over the course of a day, I'll get no notifications, and the number bubble on my Mail app icon won't change. If I open my Mail app, any and all new emails appear instantaneously and I get the notification at that moment.

In Settings>Mail>Accounts>Fetch New Data I have "Push" set to on. The relevant email account set to "Fetch." And the fetch setting is set to "Automatically."

Something interesting is that the Mail app is not on the Background App Refresh list in General settings. Should it be? I would think so, but there doesn't appear to be an "Add App" option in the Background App Refresh settings, nor is there any kind of "Background App Refresh" toggle within the Mail app settings.

Why isn't my Mail app fetching new emails or sending notifications until I open the app?

Any help or ideas would be much appreciated.