I don't understand the premise. The point of a CAPTCHA is to tell Computers and Humans Apart, that's what the CHA stands for. You cannot hope to do this test using a proof-of-work system where the work is computer work.
Call this a client rate-limiter, or whatever else, but it is obviously not a CAPTCHA and cannot function in this way.
Another obvious problem is that server hardware is vastly more powerful than the average user's device. If you set your challenge to an amount of work that doesn't meaningfully drive users away and/or drain their batteries, you are allowing a malicious server to pass your challenge tens of thousands of times an hour.
Call this a client rate-limiter, or whatever else, but it is obviously not a CAPTCHA and cannot function in this way.
Another obvious problem is that server hardware is vastly more powerful than the average user's device. If you set your challenge to an amount of work that doesn't meaningfully drive users away and/or drain their batteries, you are allowing a malicious server to pass your challenge tens of thousands of times an hour.