What benefit at what cost?
I don't know anybody around here who plants a cover crop in bean or corn fields. They might plant some beets and radishes to feed the deer that are smart enough to dig for them, but I've only seen that done in alfalfa and wheat fields and there is very little wheat grown around here these days.
Over time there are huge benefits to soil health; especially micro-biome. Plus, it keeps sediment and nutrients out of the local waterways. I'd suggest throwing some cheap clover out with the grass cover crop just to get the cheap N and sequester nutrients.
Fortunately, our farms are pretty flat and don't have any notable water erosion issues that the few permanent swales/waterways don't address. We also still have Osage Orange hedge row windbreaks on the perimeters of most of our farms that helps to prevent wind erosion.
I overseeded every bit of pasture I have clear enough to drive a tractor through.
Used the fertilizer broadcast method with a couple hundred pounds of fertilizer per acre mixed with a heavy dose of rye and oat seed. It probably cost me a bit over $100 per acre.
It rained shortly after I put it out, and last week we had 3.25" of followup rain. It's really coming in great now.
My benefits are that the cattle will have some grazing throughout the winter months as long as I get an occasional rain to maintain it, (in a horrible drought year, that means a lot, especially when round bales are fetching $150 per bale.)
Another benefit is the pasture soil condition will be better in spring for the regular green up due to the recent added fertilizer.
More and more are doing that down here.