Safari has had add to home screen for the best part of a decade. Giving web pages the ability to spam users to add it is an anti feature.
On Chrome you can trigger a add to home screen prompt from your own UI, making it a two touch experience. On iOS IIRC it only works on Safari and not in Chrome, but I could be wrong.
> no one wants notifications from random websites
I am not talking about random websites, but rather web applications that you interact with on regular basis, if Apple cared about the user experience they could have just moved the notification prompt behind a user interaction.
> shared worker ... They burn battery life
If you are building more demanding websites shared workers are a great way to share state and computing resources across and thus saving battery life. Spawing a Worker per tab is more demanding.