When I was young, straw was used almost exclusively for livestock bedding and fertilizer after it had served its use as bedding.
Balance was quite important-- too much straw on the fields yielded too much carbon and was not good for the crops. Too little defeated the purpose of using it as a fertilizer.
There was occassional trading between farmers who had too many animals and those who had too few to achieve the right mix. But far more common was to stack and burn the excess used straw in the fall. The ash reduced the carbon, converted it potash and had just the right amount of nitrogen for spreading on the fields.
It was a good combination used for generations, but I found that it applied mostly to large livestock (sheep/pig size and above).
I once did it with chicken straw. Poultry manure is so high in nitrogen, that the straw pile burned for three days. It also had to be spread a lot thinner on the crops to avoid nitrogen burning. Needless to say, burning doesn't release a pleasant odor and it is best done when the prevailing winds are blowing away from your neighbors.
I once spent a year in Cambridge, England. The local farmers used to harvest their crops—wheat, barley, corn, or whatever—and then when the stubble had dried out, they would burn it off. That quickly prepared the land for planting winter wheat, by removing the stubble and getting the remains into the ground, and I suppose as you say reduced the carbon.
It was an interesting sight, although things got pretty smokey right into town in the burning season. I don’t know if they still do it. I doubt it, because the greenhouse gas folks would probably be all over them if they did.
Gone with the fox hunts, I expect.