There really isn't any absolute right or wrong. You need to do what works best for your soil.

I would say in really tight clay, like we had in our childhood home, it was impossible to till.

In other clays, like I where I live now, tilling in organic matter has helped the soil workability such that I no longer need to till, but I did when I first starting trying to plant stuff.

And in the sandy loam where I lived previously, I never had to till. The soil still needed organic matter, but I could plant in the soil without tilling.

What Joepyeweed said. There are clay soils and then there are other clay soils, in the uphill part of my veggie garden, I still need to till stuff into the clay. Lower down the hill, the composition changes slightly, and I can easily plant stuff and lay the compost on the surface.

However, a Mantis tiller isn't going to get you too far with a heavy clay soil, at least at the beginning. You might want to rent a big, 10 hp rear-tine honker to start things off - till in a foot of stuff.

What I have found to be the best for the red clay we have is to till once and then layer after that. Good natural fertilization is also extremely helpful.

I can get down 2 feet with the Mantis if the soil is set aside as I go. How much deeper do I need to go?

TWO FEET WITH A MANTIS!!!! I think that is plenty deep.
A mantis just bounces around and scratches the surface of my soil.

Yes, but it takes a few passes to do this and then add the amendments as the Mantis fills the area back in. It isn't easy and it helps if it has been raining in the past few days before attempting it on fresh clay. It is kind of cool to see a whole bag of leaves get disintegrated into the soil and see how "puffy" it becomes while amending it.

My garden is small, so I turn soil by hand. Since I learned soil should not be tilled, didn't turn it for 3 years, only added compost before planting. This fall when I cleaned up, I decided to turn soil over and found it had compacted a lot . It was quite a bit of work to dig it up and loosen the soil. Now that I've done it I wonder what kind of soil does not need tilling. My certainly did.

Tilling any soil does disrupt the Soil Food Web. Maybe possibly a clay soil might need to have a lot of organic matter tilled in once, but my experience is that most any clay soil (there are a couple of clay types that do need tilling, once) will allow organic matter to be incorporated with no tilling. Maybe, possibly if insufficient levels of organic matter are put on soils they will not have enough humus to maintain good tilth and then they become compacted.
No matter what tilling soil is harmful. Tilling soil has no real benefit.

Till versus Turn. I can understand tilth being ruined by vigorous mechanical tilling but does turning the soil over by shovel or fork do the equivalent damage?

I have a few 3 x 8 foot raised beds and just finished incorporating two bags of old leaves and 120 pounds of composted cow manure into each. After spreading on the surface, I took a pointed shovel to the top 6 inches and turned the stuff over into the existing soil. The soil is loamy and not clayey and I have been planting there for 5 years. When I add compost, I put it on the surface and take a fork and twist it in the soil to work it into the top but do not turn the soil over onto it.

I once had a Mantis type tiller, and it just bounced around on top of the clay - If you can get down two feet, then certainly no need for anything bigger.

With the soil I have, it is best to wait until the freeze - thaw cycles all winter long have broken up and expanded the soil, then wait until it is nearly dry, and then till in as much organic matter as I can find. If I do it when its too wet, I get bricks, and if I do it when its compacted, I get smaller bricks with a bunch of organic matter in-between.

If I'm lucky, my sometimes there, sometimes not there, swarm of huge night crawlers shows up, and then all I have to do is put the organic matter on top, and they do the rest.

If I can, I fork / gently spade in organic matter in the fall, and stir it up a bit with a fork in the spring before planting. Thats what I'm doing right now, collecting leaves, grass clippings, last years compost, and so on and laying along the rows in the veggie garden, and when I've got it all spread out, I'll spade it in so the wind won't blow it all away this winter. A several week process.

The only time I even think about tilling is when I am trying to plant plugs or shrubs... anything that requires putting many holes in the ground.

For seeds, I just make a slit in the soil, then I cover with mulch and/or compost.

After three years, it is easier to make holes in the soil. But prior to that, its very difficult to dig a hole for a small 3" plug.

I do the exact same method as jimla but tilling works fine it is just a different method & you can use whatever method will produce the end result you are after. I want the soil structure of no-till & dont want to chop up the worms as they are the tillers now. Etc. I think the layering method is probably the best & least work but bringing the soil to optimum productivity takes time & constant maintainance no matter what you do. My Grandmother used to have a rich, very dark clay soil, in the hills of CA & they grew a garden every year, plowed with a mule & never had to fertilize. This was over a span of dacades.