As brought up in #883, I think we can probably simultaneously work around the issue there and improve the Zulip user experience by deferring the "allow notifications" popup until after the user has finished the tutorial (it's kinda confusing that it pops up when it does already).
While we're at it, we probably want to make the browser popup be triggered from a message that's part of the Zulip app and explains that notifications are important and the user should accept, to help ensure users don't just auto-block Zulip notifications out of habit.
I think it's definitely possible for a frontend to tell whether notifications have been denied (I'm pretty sure I've seen products display a "you turned off notifications, can you click this?" message). And https://developer.mozilla.org/en-US/docs/Web/API/Notification/permission seems to confirm you can see if they've been explicitly denied.
So a good implementation plan here would be: