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The sites I build at work have proper push notifications. This problem has already been solved.
I will never enable push notifications from websites. This problem has not been solved. It also doesn't hold across devices or if the browser isn't open.
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).