Sounds like you’re describing the Progressive Web App (PWA) install that Chrome recently added. Coming to Firefox (if not there already), and Edge. Not holding my breath for Safari though.
Mobile Safari supports PWAs and quite a bit of infra like service workers. Last time I looked it didn't work well in the background (reloads page on resume) but seems like they have most of the building blocks.
EDIT: This post [1] says background switching should be working in iOS 13.
EDIT: This post [1] says background switching should be working in iOS 13.
[1] https://medium.com/@firt/iphone-11-ipados-and-ios-13-for-pwa...