Your premise is correct.

I see nothing wrong with mixing hay (green) and straw (brown) as compost feedstocks. Having said that, old hay (as in weathered) may not be as strong a green as expected.

Alfalfa bales will heat up quite nicely by themselves with the right moisture. :-(

Lloyd

I mixed alfalfa hay with straw two years ago. I threw in some kitchen waste from time to time, but the bulk of it was straw and hay. It made GREAT compost.

You could make compost, but probably not "decent" compost. Compost is supposed to have a fairly high level of nutrients, but that depends on what materials are put into that compost. Straw would be your major ingrediant and most of the nutrients would be gone, sent to the grain the straw grew.
If the material going in the mix is low in nutrients the finished product will also be low in nutrients.

Hay may or may not make good green material. It is dependent on the protein level (N) in the hay. Some hay is more akin to straw while other hay could be up to the level of alfalfa. The round bales could even be different according to how deep the hay comes from . The outer levels will have less than the center. And this also depends on the age and how it was stored.
Some hay has been treated with herbicide. Some of these herbicides can show up in the garden and kill your veggies. (Grazon being one of them. It can even be passed on through the manure.)