It took me a while to figure out what you meant by no swing.
I will assume this demo (by Michael Weiss, not in competition)
https://www.youtube.com/watch?v=mFGLGFv8Mu0is approximately right. (I cheated, and viewed it at quarter speed, so I could see exactly what he did.)
I see the right leg starts backwards, than starts to swing forward, but by the time it comes approximately along side the other leg, that leg and torso have already begun to rotate, so the right leg never really passes the left leg - an incomplete swing. I guess that is what you mean by not swinging.
In fact, stopping that swing next to the other leg appears to be part of the way that right leg motion imparts a CCW spin to the rest of the body. Hmm. Actually, quite logical, good applied physics.
But I see that he does use a fair amount of strength for the jump itself, even on the single axel, because he gets fairly high off the ground. Which in turn means he has more time to complete the rotation and other motions.
Tonya Harding uses a lot of strength to get high in the following video, too (in competition, but it was a triple axel, so maybe height was needed):
https://www.youtube.com/watch?v=tIGoWGjetogthough she swung her leg around in front after most of the jump was complete.
I was mostly just curious about how much core stabilization mattered - because none of my coaches discussed that (they were mostly ice dancers, and talked about stabilizing the upper body instead, perhaps to imitate relatively stiff "frames" of ballroom dance), but it makes sense from a physics perspective, in terms of
1. Stabilizing the axis of rotation. (Because letting the the axis drift wastes angular momentum.)
2. Keeping as much as possible of the body mass as close as possible to the axis, minimizing moment of intertia.
3. Removing other internal motions of the body that would waste angular momentum.
4. In general, it helps efficiently transfer momentum between parts of the body (like the leg, foot, and boot, to the rest of the body). (I've gradually figured out this helps in some other sports, such as kayaking.) Because if you let some of your momentum, energy and muscle strength move the wrong joints, much of that momentum, energy, and muscle strength are wasted.
I guess my coaches were used to working with better athletes, who had already figured out that stabilizing the core matters when they were small children. So they never saw a need to explain such basic ideas. They may not even be conscious of doing it. But I was a nerd as a child, uninterested in athletics, and lacked the most basic elements of athleticism, so I needed such an explanation to get much of anywhere.
OK, I think I will drop out of this thread now. In all likelihood, though the original poster didn't include a video, my issues aren't her issues, or those of most of the people on this forum, so they are irrelevant to the thread.
In any event, I'm not ready to do any serious jumping, as I am still recovering from an injury, and I've never even done a really good single jump.
But thanks for the help, Doubletoe.
I can go back to my (mostly) waltz jumps, and see if stopping the leg swing and stabilizing the core will make them better.