I had the exact same problem, MacBook Air late 2014.
On Elementary Os Loki 4.1
after doing everything exactly the same as you, I ran:
sudo modprobe -r facetimehd
sudo modprobe -r bdc_pci
Cheese and "camera" seems to work all of a sudden. I would suspect it was from a linux kernel update or Elementary that I think happened. If this happens to you, either run it as sudo or wait until ...
AFAIK, there isn't, but I use cheese for my webcam.
sudo apt-get install cheese
It has many more options than the builtin camera. Yea, I would love to stick with the builtin stuff, but it needs many ...