Because defining what is a web site and what's an app, strikes me as particularly impractical idea. You correctly point out that yes, there are a number of powerful APIs that should be behind permissions. But there are a number of permissions already, so we need to start bundling them and also figure out how to present all this to the regular user.
Frankly, I wouldn't know where to begin with all this.
Distinguishing between site and app, e.g. via an installation process, is equivalent to a permissions dialog, except that you're now advocating for one giant permission dialog instead of fine-grained ones, which seems like a step backwards.
The new permission dialog wouldn't grant all of the finer-grained permissions - it would be a prerequisite to requesting them in the first place.
Curating known good would equate to some sort of app store. There are probably initiatives to make one for web apps, but it kind of makes me sad to think of applying that to the web, which is supposed to be a free and open commons (although I suppose Google already de facto controls enough of it to be considered a bit of a gatekeeper).
Making the user the arbiter of "known good", ie reliance on permissions dialogs, is not perfect but it's what we have. Yet I fail to see how your proposal of "just add ANOTHER dialog" improves the situation.
Many APIs should be gated behind being a web application. This itself could be a permission dialog already, with a big warning that this enables tracking and "no reputable web site will ask for it unless it is clear why this permission is needed - in doubt, choose no".
Collect opt-in telemetry. Web sites that claim to be a web application but keep getting denied can then be reclassified as hostile web sites, at which point they not only lose the ability to annoy users with web app permission prompts, but also other privileges that web sites don't need.