- Gmail - Calendar - Online support chats where I get a notification when an agent becomes available/responds so I don't forget about the tab - A custom tool I wrote for myself (web notifications are the easiest way to get push notifications onto your phone without having to go through the overhead of full app development)
I bet there are some people who genuinely like the idea of a news site pushing them the latest news. I doubt they outweigh the number who accidentally enable those notifications and then can't turn them off, but I haven't seen user studies and I could be wrong; average users often have habits that seem weird to me.
I have, however, also seen what really abusive notifications can do: A friend enabled notifications from a shady site or the ads served on the shady site (because users don't understand dialogs and end up clicking a random button, so ask them often enough and one of the clicks will land on "yes"), and now their Chromebook was telling them that they have a virus and need to {install a malicious extension,give the scammer money,etc.}. Since then, disabling notifications is on my mental "browser hardening for when I set up a browser for inexperienced users" checklist.
I'd wholeheartedly support blocking the notification dialog for all sites, limiting notifications to a barely-noticeable icon that the user needs to actively click. Much harder to convince a user to take an action than to convince a user to click a random button to get rid of a dialog that's getting in their face.
Also, an easy browser-controlled way to unsubscribe from notifications, and enforcement based on the rates of a) people clicking "no" in the permission dialog, b) unsubscribe rates. If the metrics show that your users aren't happy with your notifications, no more notifications for you.
I've never turned them on for news types sites though.