Lack of water in the woods maybe as it appears to have been cleaned out. All pertinent leaf litter removed. All water drains down the hill..........yes, and ends up next to the house! The turf is green! Unless the soil is tested to make sure and a divider or retainer enclosure is made at the drip line of those trees, then this hillside will not improve.
I can guarantee that the tree roots were severed at the time of construction, so there is nothing but roots down on that slope. Aerating is impossible!!!at first glance. You may only get down 2 inches at best. Build a divider.........add more soil, resod that hill. Plant some native plants to slow down or divert water. Anything to reduce soil and silt loss on that slope will improve turfgrass growth.