When I'm in one game, and I get a notification from a second one, clicking the notification banner at the top should pop me into the second game. That would make it smoother than exiting to my game menu, and the looking at the list of games I have and trying to remember which one the notification came from.

I was really hoping this was a suggestion to fix how the that notifications cover up other crucial information in the game.

I mean they only cover it up for a few seconds, and you can tap them to make them go away.