There is a big problem with your theory.
Yes the depletion of much needed proteins is one side affect that is if one feeds a low protein scratch.
Chickens need just as much proteins in the winter as they do in the summer, the lack of protein in the winter mean that the chances are good that there eating more feed, getting thin, and having trouble regulating there body heat (proteins produces more heat than a grain). There can be a list and a half of problem that poultry can get when forced to eat a low protein diet.
Another problem with feeding a lot of low protein scratch is that it is also high in energy, so now your birds are being fed a low protein high energy diet which can make the birds fat, poor layer, unhealthy and since chicken eat to fill a caloric need the could be not getting enough nutrition.
Have you ever done the math to figure out just how poor of a diet that is?
Here is a good example, if your feeding a average low protein layer feed say 16% protein and a average low protein scratch say 8% protein at around 50:50 by weight you would be feeding a 12% protein feed which is way low for the average backyard chicken even in the summer.