Your information regarding push seem outdated - both statements are NOT true. Since iOS 16.4 you can deliver push notifications even when the PWA is not actively in use. You need, however, to have the website added to the home screen (=PWA install in iOS world). I use both features on lockmeout.online - which core functionality is based on delivering push notifications on the lock screen. You need however to give permission/configure lockscreen notifications as with any other (non-PWA) app.