>> Here at our house we have begun bagging our clippings because after not doing it for a year or two, our lawn was getting so congested with thatch
When I was a kid, doing the mowing, I fought to leave the clippings in place. But the one spot where the lawn was lush (septic drainage field) did accumulate thatch. So I would wiat a few days for the cuttings disapear or turn brown, then rake the thathy area, spreading the clippings around.
Then I learned to mow so that the mower would throw the cuttings from the lush area onto the sparse area.
Now, I would take the bags of clippings while still green and rich in N,and mix them with 3-4 parts of paper, sawdust or other "browns". That way I would get 4-5 times as much compost from them. Whether that compost goes back onto the lawn, or to a garden bed, depends on what needs compost the most.
Does anything compost faster than grass clippings?