Some games benefit from push notifications as reminders to do something like "feed my dragon or else it will die" or "your free spins are available now". These are integral to the core mechanics of the gameplay vs. a marketing message like "buy some coins for $1.99, on sale now!" or something to that effect.
Waiting by not playing a game is never a "core mechanic" of a game. It's either an excuse for the app to grab your attention (play now to get a bonus) or a way to drive IAP (pay so you don't have to wait).
I'm thinking of something like a Tamogotchi where you need to take action at specified intervals and push notifications would seem like something I would like as a reminder.
