Correct me if I’m wrong but isn’t one of the strongest signals recaptcha uses to prove “I am not a robot” the fact that you’re currently logged into a google account, have the associated cookies, etc? This “phoning home” from all these sites is a clear privacy challenge.
The good news is Safari and Firefox are making this harder and harder. The bad news is that you’ll be solving a lot more captchas.
I wonder if privacy conscious browsers making their users have to solve more captchas will lead to more people disabling the privacy features... or there will be a time when captchas may become micro transaction based, so for a fee you might be able to skip them. A browser with support for an anonymous cryptocurrency to make the payments should be private and easy to use enough, but it might become an interesting world where you need to pay to remain anonymous.
The good news is Safari and Firefox are making this harder and harder. The bad news is that you’ll be solving a lot more captchas.