The sites I build at work have proper push notifications. This problem has already been solved.
App PNs also don't "hold" across devices. You don't need to have the browser "open", whatever that means.
BTW there are tons of apps that do push notifications wrong. There are many apps which keep sending you PNs after you log out or switch accounts. This can be really sensitive content (e.g. chat apps).
4.2 Minimum Functionality
Your app should include features, content, and UI that elevate it beyond a repackaged website. If your app is not particularly useful, unique, or “app-like,” it doesn’t belong on the App Store.
The important part is that they go through the app store, so that apple has complete control of what goes through.