I've implemented PAT on a service. A server would only want to require PAT when it wants to see if it is dealing with a human. In the context of, say a blog, you would require PAT when someone wants to make an anonymous comment or create an account.

For requests which are "reading" you just serve content, unless for some reason you only want human eyeballs to see your content.

In your proposed scenario, reading the publicly accessible contents of the web, there should be no problems. (Of course some percentage of sites will accidentally have required PAT at any time and be unscannable, but presumably they figure that out and fix it.)

Now for the good side: I, reluctantly, implemented a geolocation filter to control anonymous content additions to that service I was alluding to. I felt bad about it, but I also felt bad having to filter out content spam every day. It turned out that all my strange content spam came from one country, so I banned 143 million people from anonymous content creation for my convenience.

With PAT I can remove the national ban and let any "probably human" in.