I have read that the white dutch clover mentioned... fixes at least 300lbs of nitrogen per acre per year according to the USDA.
White clover also is a good source of calcium... great for egg laying
chickens.
The leaves and flowers are also edible for humans.
Then all of the pollinators and beneficial insects love the stuff too obviously.
So... with all that being said... I totally top seeded my entire lawn around my house with the stuff. I found a 9lb bag on Amazon for cheap. They were already inoculated with the nitrogen fixing bacteria too. It's beginning to come up now. I can wait to see how the nitrogen hog lawn reacts to natural fertilizer.
I have read that alfalfa fixes nitrogen as well... and has roots that grow up to 15ft long. Sounds like a hardy plant!
Every time you mow you will have
root dieback... and the lawn will get fertilized.