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.
Yes and...no (at least in my opinion). I agree completely that you can cut with a sword and also thrust with a sword. I also agree that you can make a thrusting movement with a jo but also a cutting movement as well. But the way i see it there is an important difference.
When one cuts with a sword he tends to "draw" a part of a circle with the tip of his sword, because the sword has a curve and the shoulders must extend and withdraw again. But one makes a shomen uchi with the jo the tip makes also a circle but the jo remains straight and the feeling one gets is not that he is cutting but hitting someone.
When one thrusts with sword the thrusting movement starts from the tanden and extends forward, with the feeling of penetration.But when one is thrusting with a jo, he is altering his hanmi slightly turning the hip and the foot towards the inside and makes the move with the feeling of hitting again.
So when we thrust or cut with a sword the feeling is one of penetration,and when we make a tsuki or a shomen with a jo the feeling is one of hitting.We could say that the sword is a cutting weapon that could also thrust and the jo is a hitting weapon that could also "cut".
Both weapons are teaching us the principles and movement of aikido, but i personaly prefer to take as reference the sword for a shomen and the jo for a tsuki.
Yet this is my personal view,others may do differently...