You would need to trust the service the same like people are trusting logless VPN providers with their sensitive internet traffic logs.
Even if the footage was never recorded or deleted instantly after. Even if it was 100% anonymous and 100% guaranteed secure, it still feels like some creepy dystopian nightmare.
A system like this means we've catastrophically failed with technology. It's too dehumanizing. At this point, burn it all to the ground, we're done.
If webcam or mic is not detected and webcam recording or mic recording was given then obviously malicious actor is trying to cheat the captcha and submit fake recording/s.
Edit: You would be prompted with something like: "Wave your hand from left to right." And then video recorder would be presented to you and you would record yourself doing it and review the footage and submit it to the captcha task.
The same applies to mic and voice recording. For example you would be prompted with: Please say: "Life is like riding a bicycle. To keep your balance, you must keep moving." And then voice recorder would be presented to you and you would say it, review it and submit it in order to fulfill captcha task.
I'm betting on a physical recording device that a person has not on the ability or inability of bots or an AI to come up with a human video recording or a voice recording.
A bigger problem is that once you've given some random site permission to access your mic or camera, it can use that access for anything it wants.
And speaking of recordings you can always check metadata and see if it is genuine or fake(edited).
Metadata is trivial to modify, e.g. FFMpeg can do it, and it usually doesn't say anything about genuine-ness anyway. Besides, if you're generating your fake input in software, you'd just generate metadata to go with it.
Sorry, but there are so many holes in your idea that it seems completely unworkable. Even if new developments made it more feasible (new webcams and microphones that cryptographically sign their recordings? Still fakeable by the speakers and screen setup above. Depth-sensing cameras might be trickier.) people would rebel against allowing any and every site to access their webcam and microphone (and hence possibly spy on them).
This seems even worse to me. Wave at the camera to prove you are human... This is a whole other level of creepy above depressing captchas. I would not do this, ever.