Rami, welcome to the forum.
It is hard to advise someone on how to proceed in such a large area. I want to say plant mushrooms though that is not really an option for large areas.
This post may offer some understanding as to what the problem is with soil health:
Bryant said, "This however won't directly accelerate the remediation process because that process is interdependent upon bacteria, fungi, amoeba, nematodes and other microorganisms all building their microbiota world and thriving.
While root systems can help with the overall soil health by increasing the ability for water infiltration and air exchange, the roots are more of a third stepping stone in this particular scenario in that the roots aren't really necessary for the process to proceed at any pace.
If you increase the bacteria and fungi, the other microorganisms will increase because of the increase of food supply, so the only fungi that would really benefit from more roots would be the mycorrhizae, which are not actually part of the remediation battalion.
https://permies.com/t/93813/Recovering-yard-rounduped-helpful-neighbor#768385
If I had this problem I would look at other areas on my property where the alfalfa was not grown or maybe even consider container gardening for a few years.
What you have been doing by cutting the alfalfa and letting it return to the soil maybe you best option.