Don't worry too much

. You do not need to take everything back out. Get some of your material composting well, you want to get all the microorganisms going strong and the composting process aids greatly in doing that. Then when you have some good
compost made up, make some compost extract (like compost tea but without the aeration and growing time), and use the extract to water the area. This will introduce loads of the beneficial microorganisms all around the yard.
Getting a fungal dominated soil food web will work against the grasses and support more desirable plants.
Dr. Elaine Ingham explains it all wonderfully in a series of lectures on Youtube "Common Ground".