My guess is when they are talking "hay" they are assuming at least a grass/legume mix of some sort. While the soil benefits greatly from the OM of both the root systems of the hay and also benefits from the thatch layer of the dead plant material that accumulates on top of the soil, that is not the only reason to include "hay" in that rotation. But, a grass mono culture is not much better than corn from a soil N perspective. The natural N produced by the legumes is a benefit that is known but hard to quantify. One can measure the amount of N that a clover crop will put into the soil, but the type of N is also beneficial to the soil in the fact that plant produced N does not increase salts, chlorides, or any of the other "harsh" chemicals that one finds in synthetic N fertilizers. Almost every type of synthetic N fertilizer will lower your ph to some degree, some more than others. I have a chart stored in my notes that lists the amount that each type will decrease ph, if I find it I will post it sometime. The natural N from the legume roots is far less harsh on the soil and doesn't "fry" some soil life like synthetic N will do. Google N fertilizer and earthworms, you will see what I mean.
Overseeding rye will serve to add to your OM no doubt, and the rye may even "mine" small amounts of N from deep in the soil profile, but it cannot add N to the soil like clover or alfalfa, thus when you grow your next crop you will be forced into adding more synthetic N to the soil and thus lowering your ph and potentially killing off beneficial organisms in the soil.
