I don’t understand why this would be unrealistic with the fully-confined backyard model (good term, btw.) Unless someone is gone 14 hours a day, it’s easy to provide a basic whole-grain meal, dry or fermented, plus varying additions in late afternoon, etc. Feeders could be easily maintained. If the run is properly predator-proofed, it would have the added benefit of not feeding the local wildlife as well.
In fact, Perris’s feeding model would also work in this setting quite well.
It still wouldn’t have the full range of nutrients that foraging supplies, and it sounds miserable for the birds, of course. But it’s probably similar to how chickens are kept in severe-winter areas where there’s just not much left to forage after a while.