What native trees (of Kansas) can be cut right now and rooted outside in the soil with some success without much care except water and amended soil?
Willow. That's about it.
Red Mulberry and Osage Orange might work, but I've had almost no luck rooting cuttings of those just by sticking 'em in the ground, as you've described.
I've had mild success rooting cuttings of some of the native plums, just by sticking them in the ground.
I don't know about cutting the roots and sticking them in the ground. I don't think any tree accept the Tree of Heaven can be started that way, but the Tree of Heaven is an invasive species and should not be grown at all for any reason.
However, I know that Silver Maple, Elm, and Sycamore all transplant very well.
There is a very good book "Landscaping with Native Trees" by Guy Steinberg & Jim Wilson... ( I met Guy since he is local) use that to pick out the tree(s) then we can tell you the easest way to ppropagate
BTW, using the proper medium to root cuttings is fine, but when planting your trees out, be sure to remove as much of the differing soil type as possible (some even suggest bare-rooting woody plants). Woody plants do best without amended soil.
Here is a link that might be useful: Planting a Tree or Shrub