I think it's difference in light exposure being the determining factor in how many eggs are laid, and not food type.
I have an open air coop covered by a clear tarp, so my chickens get all the light available all the time. There are mature trees nearby that provide some shade depending on where the sun is in the sky, but now that the leaves are gone, shade is minimal. They have clutter in their run to hide underneath, but in general are exposed to whatever the sky is doing light-wise. I've noticed that if we get a few really overcast days in a row, the number of eggs I get decreases. When it's sunny again for a few days, production goes back up.
I have a mixed flock, all 8-9 months old, and my production reds haven't stopped giving me one egg a day, but my olive egger (who lays brown), BO and SGEs might skip a day or two if it's been particularly cloudy for a while. Right now we're getting 10 hrs or less of light a day, and I'm in north Alabama, but I think it's applicable.
I feed Purina all flock with oystershell on the side and ~10% food scraps or scratch.