Can't you just use ServiceWorkers in a regular webpage? Seems like you can. That would allow you to have an offline persistent website while still using a webpage. https://caniuse.com/#feat=serviceworkers
For example, we built https://usebx.com/app with plain old HTML, JavaScript and CSS. Yet, it can be installed on your phone, send you push notifications and even works offline. So I would categorise it as a PWA rather than a website.
My point is, if you're using service workers, you're more a PWA than a standard website.
(but I also avoid using the term for exactly this reason)