Yes
Would you like notifications when someone mentions or DMs you in your work chat?
Yes
Would you like to know when you get emails?
Yes
Also only one of the things you mentioned has anything to do with notifications.
The rest of them, definitely not. Interruptions are extraordinarily costly. I'll respond within a few hours and when I message you about something, I have the same expectations.
The browser is free to refuse every notification request AND make websites believe the notifications have been enabled. Unless smartphone OS vendors are kind enough, you can't do that with app notifications.
It seems you're arguing against the concept of push notifications itself, not whether the native or web app implementation is better.