Well, apps can only "run in the background" for up to 10 minutes. So sure, an app can generate a notification popup itself, but it's pretty limited as it won't

Well, apps can only "run in the background" for up to 10 minutes. So sure, an app can generate a notification popup itself, but it's pretty limited as it won't be able to generate a notification after being backgrounded for more than 10 minutes. And the 10 minutes is also only if the app is designed to extend the duration as long as possible. Normally it would get cut off after 1 minute. So because of this it seems that in the vast, vast majority of cases apps choose to send their notifications from the Apple Push notification service.