I have some seriously thick/hard/compacted and tree riddled yard with reddish colored clay soil. I tried to shovel it while it was dry and almost killed myself without getting so much as an inch

Use a pick to break up the dry stuff and a shovel to move the chunks around.

However, if you just layer the organic matter on top, the soil underneath will slowly improve. Make lasagna beds and plant through the beds and skip the tilling.

Check the tree species for their tolerance to having the soil level changed around them. Some are tolerant, others go into a decline and die.

If you are like me, the first 15 minutes with the pick you will feel like you are accomplishing something. The next 10 minutes will feel like work. The next 5 minutes you'll feel like you're dying.

It's too much work for moi, but give it a shot and let us know how you get on. Me, I would opt for about a foot of sawdust mulch. 3 seasons later the worms will have done your work for you (and it will stop looking like sawdust 3/4 way through season 1. For faster results, plant something in it like alfalfa or buckwheat).

YMMV

In an effort to convince son-in-law that the leaves from the trees was a really good thing to add to the soil, instead of raking, bagging, and throwing them away, I piled some leaves on the clay soil they have there around Columbus, Ohio one fall and the following spring he found that what had been a hard soil that a garden fork would not penetrate was now a loose, friable soil that the garden fork would, with no effort at all, penetrate to the handle. Lee Reich, in several articles he wrote in Fine Gardening, has strongly urged people to consider making soil like Ma Nature does, from the top down. Add your organic matter to the soil by piling it on the soil and letting the Soil Food Web move that into your soil for you. Much easier then any tilling you might do and much better for the soil since any tilling will harm the Soil Food Web.