It's a symptom of the whole "we converted our document platform into an application platform" debacle that typifies the modern web.
Notifications make no sense for the majority of websites, but if you use, say, a web-based email client, then you probably do want them.
Yes and no. Prompting for it modally the way they do now is for sure wild, but for some webapps (e.g. Slack) it makes plenty of sense. I think Firefox used to have a UI they used for some things where they'd inject a non-modal bar with a couple of buttons inside the content area. This sounds like the right type of UI, maybe at the bottom of the viewport.
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