I hate that a regular website I may tap on a search result would ever want to send me notifications. But, if a site that is clearly more of an “application” to me (like an online spreadsheet or chat app, etc) wants to send me notifications, that could be a useful feature.
If there were a more clear delineation between the two, I could just avoid tapping anything that is labeled as an “app” in search results (or just ban them from my search engine), and restrict myself to “plain old webpages” only when searching. And “plain old webpages” would be even further restricted than they already are: No notifications prompts, no ability to see my accelerometer, no offline storage available, etc etc.
It’s nice that the web has turned into a viable app platform, but it doesn’t mean I want every page I visit to be allowed to have “app-like” capabilities. In an ideal world 95% of the web would just be plain old documents accessed over HTTP(S) and the “apps” would be distinguishable as such.