Notifications have fine tuned controls on both iOS and Android with schedule Do Not Disturb features and varying levels of priority (iOS has 3 tiers). Plus per-app controls to disable them from annoying apps, which in newer versions of iOS prompt you before they turn on, making them opt-in.
It's essential for communication apps (texting, email) and useful for breaking news or a simple highlight of big news on WSJ/NYT on your activity feed, so you don't need to open either app to stay on top of big events if you're a news junkie like me.
I'd personally go as far as saying push-notifications + the activity feed is very underrated feature of smart phones and has become a core part of people's day-to-day lives (which plenty of technologies can't say).