I say this all with no solution to put forth. I'm betting users would not be happy having to assign a priority level to each app and furthermore not all notifications from an app are of equal importance (ie. IFTTT SMS vs iMessage from my Mom).
One thing, unrelated, that I hate about notifications (that IMHo greatly diminishes my use of them in iOS) is the fact that in the notification center they are grouped by app which is one of the stupidest things I've seen in iOS. I want my notifications weaved together and sorted by time not by app THEN by time. Under iOS current implementation you NC could look like this:
* GMAIL
* 1 min ago - Blah blah blah
* 5 days ago - Yada yada yada
* 10 days ago - Bippity boppity boo
* MESSAGES
* 2 min ago - URGENT MESSAGE
* 3 min ago - ANOTHER URGENT MESSAGE
That's just stupid, it should be: * GMAIL - 1 min ago - Blah blah blah
* MESSAGES - 2 min ago - URGENT MESSAGE
* MESSAGES - 3 min ago - ANOTHER URGENT MESSAGE
* GMAIL - 5 days ago - Yada yada yada
* GMAIL - 10 days ago - Bippity boppity boo
This is just a simple example but throw a few more apps in there and it's not unheard of to have something that happened 1-2 minutes ago buried a few swipes down to get past other app's notifications.