Space is and can be an issue. I'll throw this out to you. A broiler only takes 8 weeks to reach market weight. Basically, if you have a couple of additional chickens, in a couple of months, you are done with them... into the freezer!
My method is based on Joel Salatin. I have pens, each are 8'x 10', and they are moved over my pasture grass each day. They still get full feed from grain etc, but the added grass really enhances the health etc. The birds are always on fresh grass every 24 hrs. You could do the same on a small scale in the backyard. Running them over the grass doesn't tear up the grass, and it keeps bugs down, and adds fertilizer. In my pens I have about 40 or so birds. At each days end, there is a square, and by the 8th day, the grass is completley relplenished, to the point you can't even tell the birds had been thru there! With a 4' square pen or even smaller, you could run it all over your back yard and have all the birds you ever wanted to eat, and you would never overgraze your yard. The birds grow so fast, and are so young that they never even get old enough to make noise ie roosters etc.
pasture tracks from my pens in the pasture look like this after one day:
and then like this a week later:
it would be an interesting process to see what you could do in a backyard type of setting....