Proprietary web browsers are in a really good position to do something like this, especially if they offer a free VPN. The browser would connect to the "VPN servers", but it would be just to signal that this browser instance has an internet connection, while the requests are just proxied through another browser user.
That way the company that owns this browser gets a free network of residential IP address ready to make requests (in background) using a real web browser instance. If one of those background requests requires a CAPTCHA, they can just show it to the real user, e.g. the real user visits a Google page and they see a Cloudflare CAPTCHA, but that CAPTCHA is actually from one of the background requests (while lying in its UI and still showing the user a Google URL in the address bar).
At the limit, this problem is the problem of "keeping secrets while not keeping secrets" and is unsolvable. If you've shared your site content to one entity you cannot control, you cannot control where your site content goes from there (technologically; the law is a different question).