I have the same problem and I see a multitude of others do as well. After reading through so many posts, there seems to be some variations on the problem. In my case, the Big Sur upgrade removed Facetime from the camera's allowed apps list (System Preferences/Security and Privacy) so I can't check its box. Additionally, there is no "+" available there for me to add the Facetime app back in (this is true for other services as well). I have tried all the suggested remedies such as resetting NVRAM/PRAM/SMC with no success.
MacOS sees my camera. It works in Zoom. I suspect it would work for me in Facetime if I could ADD Facetime back in. Apple has screwed something up here and is not being forthcoming about the existence of the problem nor seems to be actively addressing it. Given the number of people who are dependent on remote connection at this time of pandemic, that is just pathetic.
I am including my system info here for those collecting data. I have placed feedback for MacBook Pro in Apple's feedback system. APPLE, WAKE UP!!
FaceTime HD Camera (Built-in):
Model ID: UVC Camera VendorID_1452 ProductID_34304
Unique ID: 0x1410000005ac8600
Hardware Overview:
Model Name: MacBook Pro
Model Identifier: MacBookPro14,2
Processor Name: Dual-Core Intel Core i5
Processor Speed: 3.1 GHz
Number of Processors: 1
Total Number of Cores: 2
L2 Cache (per Core): 256 KB
L3 Cache: 4 MB
Hyper-Threading Technology: Enabled
Memory: 8 GB
System Firmware Version: 429.80.1.0.0
SMC Version (system): 2.44f6