Just a workaround, but someone made a helping pop-up: https://github.com/ryxxn/pwa-install-prompt

Last time I checked, push messages did not work when the web app was not running, has that changed? Because I think that is the most interesting feature for users: To notify about important things while not using the app. It has also always been abused by app creators with useless stuff to drive engagement etc. I think I never allowed a single webpage to send me notifications - if for example I want to know about the latest news about plant pottery, I will probably just visit that website again...no need to notify me.