Yes they would. As the dead grass decays it recycles nutrients back to the soil microbes.

We live in a "treed" area and have live oak trees. Their leaves are about 3/4 inch by 1 1/4 inch. The trees are evergreen, but every spring they drop about 10-30% of their leaves. They are small enough to weasel themselves down into the turf and can even smother the grass. Thus I blow the first layer, before the grass starts growing, into the flower beds with a leaf blower. Then I collect the neighbors bags of leaves and make compost. Over the course of a spring, I believe there are about 10,000 bags of leaves in my neighborhood that go to the landfill. Wish I could have them all.

That depends on the care those lawns receive. If the soil is dosed with synthetic "fertilizers" then the Soil Food Web is adversly affected and probably will not digest that thatch and it will simply accumulate on the soil presenting a barrier to both moisture and nutrients.
In a good healthy lawn 1/2 inch of thatch is good and the Soil Food Web in the soil will help maintain that 1/2 inch of thatch, digesting whatever else is put there that is good for the soil.