Just to throw this into the mix as really it's my main personal question or maybe not even a question really but view. Why the sword?
Using tsuki as your example I equate that more with the spear.(Thrusts)
Tegatana 'cuts' with the sword. Circular with the jo.
A sword still has within its design the point of a spear. There are applications in which the sword is
a spear. A knife, too. Same thing. The tsuki is a spiral thrust and can be executed with a spear, sword, knife or even ball point pen. It's the same. And even effective movements that may appear externally linear are still spirals.
Yes, a knifehand strike (tegatana) can be used to cut, but also to strike, and all are spirals.
The sword is generally meant to cut or thrust into soft areas. The jo strikes hard areas - but can also thrust into soft areas. And even though the targets may be slightly different, the movements are exactly the same. A jo is also a spear. A jo is also a sword, because the power sent through the jo can cut through - it doesn't just stop. It's all the same thing.
Tsuki with the hand, sword, jo, McDonald's french fry, chopstick, fork, flashlight, toothpick, tv remote - is all the same. And it is also the same movement within gote kaeshi, iriminage, and tenchinage. All these tsuki movements operate through the front/back direction of the body. Not only uke's, but nage's too. It's the same.