My brand new M1 MacBook Air has the exact same issue. FaceTime Calls, Photo Booth previews, Zoom Meetings, recordings - everything has a flickering self-view thanks to the camera.
I followed all suggested support steps, chatted with and called Apple Support at least 4 times, followed all their instructions to diagnose, and even reinstall the OS. Finally both the Support Advisor and I gave up, and I was told to go to the Authorised Service Centre.
Upon going there, the technician found that this is not an issue with my MacBook device only, but with all M1 MacBooks. He even showed me at least 3 demo pieces in his adjacent store, and all 3 have the same issue. What's surprising is that older MacBooks did not have this issue at all. It is presumably something to do with the changes made to the camera in the M1 MacBooks.
I was told it is a hardware issue, and that he could not do anything from his end to rectify the problem. I am thus constrained to join the many others in this forum to raise this issue to Apple. Please solve this issue on our existing devices, or replace our devices at the earliest. And please realise that it is extremely unfair on all of us M1 MacBook buyers to be stuck with such a pathetic camera experience after spending so much time, effort and money to get the latest Apple device.