I have not found a good use case for browser notifications, to the point that I have turned them off completely due to endless prompts for permission from every website to push them.
(I very rarely do this. I think the only thing I have pulled out like that currently is Puzzmo, which doesn't even do notifications.)
These chats are going to last < 1 hour typically. There are times when the custom support agent needs to confer with someone else, and then will respond back. The end user, in that case, does not want to sit there with their phone open for 15 minutes, waiting for a response.
While we have an app those end users could download and get notifications on, no one wants to do that for a one off session (not even an App Clip).
Trying to get iOS users to add it to their home screen, is basically a no-go too.
So, it would be really nice if we had a way to send push notifications, even if it was time limited (allow notifications for the next 1 hour).
Don't get me wrong, Push Notification can and are abused. But there are valid use cases.