Welcome beerwolfe.
I don’t think that soil is bringing much to the party, as you notice it is lifeless, often the case with other peoples’ waste soil which has been processed and hanging around. I suspect the buried lawn soil is better quality.
However its there now, compost will help bring microbes, a little animal manure, even only half decomposed, will be worthwhile, and any of your own compost.
Perhaps soaking some compost in water and watering the liquid will speed things up. True compost tea is made by adding only a small amount of top quality compost to water, with some sweet such as molasses, then aerating and stirring for 12-24 hours, it needs special kit.
