> Maybe I'm missing something here, but it feels like you don't need some fancy realtime queue service for most push notifications, so why not use something that offloads the checks to the client?

Depending on the use case, you may well depend on the real-time aspect of push notifications. If my door bell rang while using headphones, I'd want to know immediately. I'd also be pretty pissed if a messenger took more than 5 seconds between sending and receiving on the other end, event if "closed" or in background.