I experimented last year with black cow and alfalfa on the front and side of my yard. I ripped out the old lawn in those areas and put down black cow, mixed in a lot of alfalfa pellets and then put down the sod. Both areas are very dark green and almost weed free this year. In other areas I spread out black cow and watered it into the lawn. Then applied alfalfa too. Those areas look nice too. I used CGM in March and had no problems with crab grass this year.
Alfalfa and black cow seem to work well so give it a shot.

Black Kow is composted cow manure, not much vegetative waste is added when it is piled up. This also comes from very large dairy operations, most likely not our vision of the bucoluc farms of the 1950's but rather the Confined Animal Feeding Operations types of factories of today.
If you don't mind adding material that could well have relatively large amounts of antibiotics it may be some help.