I raise both chickens and quail and determined that for me, living alone with my dogs and flocks, that a handful of chickens for the sheer enjoyment of having them with the bonus of some of their eggs to eat and 16 adult quail (for breeding) and their offspring (for meat) provides me with close to 30 pounds of meat per month, and 15 dozen quail eggs to eat. I love that I can go from hatch to the table in six to eight weeks with quail. The rest of the eggs are incubated to replenish the meat birds and a couple of times annually to replace my breeding stock. For me, this is totally sustainable. Costs me one 40kg bag of game bird feed per month ($23), and supplements both my diet, and that of my dogs who adore quail eggs served hard boiled, shell and all.
I do still have a carbon footprint, in that I enjoy beef and pork as well, and I do not raise them. If I had to live on nothing but what my flock provides, I could easily increase the number of breeders. I currently only use one of my three incubators, and have the other two as backup, so increasing the production wouldn't be an issue UNLESS the grid went down for more than a day. I have my incubator hooked up to my UPS (Uninterruptible Power Supply), and it's rated to power my computer, monitor, speakers, and modem for several hours, with the computer disconnected, it can keep the incubator going for over a day.
I really enjoyed your post, and the question presented, and agree that raising CHICKENS to feed a family of four ALL of their protein needs, given the much longer time from hatch to production (eggs or meat) would require a mind boggling number of chickens.