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).
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.