Excuse me, I didn't mention any maths, and I'm not the engineer in this thread (it's the original poster). I took pains to state that I'm not a coach nor a high-level skater - whatever I said is how my own coach explains things when we're working on doubles - so it's not like there is a need to put me in my place.
I don't think we are on the same page about what the poster was asking. Yes, change (or not) in rotational axis is a characteristic that makes Loop/Flip/Lutz mechanically similar compared to Axel/Toe/Sal, but the poster is not a skater (not sure if they actually can tell between the six jumps) and was asking if "one leg pushes upwards and the other generates spins". I interpreted this as being a question whether each leg is solely responsible for either height or rotation, so decided to address it by explaining how it doesn't hold true neither for edge jumps (with the entry edge rotation) nor toe jumps (with the weight transfer from skating to picking leg). If you do axel/toe/sal then the swingthrough and mid-air shift to landing leg contributes significantly to rotation, so the question could have also been addressed that way.
Some skaters prefer no-axis-change jumps over axis-change jumps (me included), others prefer edge jumps over toe jumps. What characteristic you find the most salient is rather subjective. Some coaches teach kids loop before toeloop or vice versa; approaches also differ on how they communicate the concept of pre-rotation (or lack thereof).
@kolyadafan2002 sorry to bring you into this, but I noticed you have a knack for explaining skating technique in words

We are going on a tangent that may be of little interest to the OP, but nevertheless would you be able to weigh in on whether I committed a cardinal sin in my explanation or not?
What you said wasn't wrong, albeit a simplification. But I would say the same about the person who replied to you - although, it seems like a strange reply in the sense that the mechanics of figure skating can fill multiple books, yet the responder took your small paragraph as you trying to describe the complete mechanics of the sport.
Anyways, from what I've gathered, your intention was to inform the OP that we use both legs to generate height and spin on jumps - which is true.
On every jump, often you first have the build the height before pulling in - using a combination of the free leg and the standing leg, which you mentioned. For many jumps (e.g salchow, loop) this involves hitting a h position, driving the knee up whilst also building height from standing leg. For some people with some jumps, this doesn't happen (e.g most 4T's, Kolyada's 4Lz). For some people, it happens on every jump. It can happen to a lesser extent or greater depending on how quickly you pull into rotation etc.
As for weight transfer.... ahhh... I think you could get a group of skaters to have a heated debate all day about this, and they would likely all have a point for their argument.
It's hard to simply jumps to a single technique, as almost everybody does something different. In this way, even though as a coach when explaining jumps I'd group Salchow, Toe, Axel as weight transfer and Loop,Flip,Lutz as not, many people do weight transfer to some extent in every jump. With quad Loop, Hanyu drives up his leg a huge amount then closes in on the axis. I would definitely say there is weight transfer there. For many people there is a weight transfer in F/Lz - albeit less obvious. They do move their body into a different position from the takeoff though. and those who pre-rotate in these jumps definitely have weight transfer. The body is used in every jump, whether to a lesser extent or higher extent.
I want to stress, using the body TO jump is not ideal (And by that I mean relying on body momentum/swinging it through) - what it should be used for, is to allow the freedom of movement in the correct way from legs and arms to generate the most height and rotation possible without compromising the stability, before tightening up and trying to minimise the moment of inertia. It should also be used to ensure a strong and stable axis, whether that involves keeping it straight on the axis or leaning forward can depend from technique or skaters build. Even that paragraph from me is a simplification though, so don't read too much into it.
On the topic of teaching toeloop - I pretty much have to teach it as the 3rd jump after waltz jump and salchow, purely as that's the Learn To Skate syllabus. If I could write my own syllabus, I'd actually teach it at least after loop and flip, as it tends to be the one where with beginners bad technique is prevalence (toe-waltz), and one of the jumps that are most conceptual. (usually most of them naturally trying to toe-waltz, and other coaches allowing them to do so). The only one I'd really teach after toeloop is Lutz - purely as having a really strong outside edge is a pre-requisite to good lutz.