Is the "Twitter PWA" something other than "www daht twitter dot calm" in a web browser?
The same. But you can add to home screen, so it feels like an app, kind of. It's pretty bad, but honestly I want social media apps to be bad to mitigate the addictiveness they build in.
notifications are an opt in feature, just like they are for native apps. If you don't want them you turn them off or don't opt in in the first place. Me, I want them on for private messages only. If twitter didn't allow that i'd turn them off. But apps like slack or at least apps like whatsapp etc would be kind of useless without push notifications.