"From the Association newsletter and other sources, I think T'ai Chi use of the term "waist" refers to the soft tissues around the lumbar spine and explicitly does not include the kua, "hips," or upper spine."
Neither am I a speaker of Chinese, but I've heard that "waist" is usually translated as "yao", but the Chinese idea of "yao" is not exactly the same as the Western idea of "waist." Then, of course, there's the "kua" or hip/upper-outside of thigh and "dang" or crotch. Taken together, they can represent the center of the body. Imho, it's possible to stand without moving thse elements, but it's very difficult to move without using them. OK, arm movements are possible, but I'm still convinced about leg movement --let alone body movement. Imho.
You also wrote:
"Steve, you mentioned treating kicks like exaggerated steps."
Yes, I used the "walking" analogy. You're absolutely right: I don't think of "waist" movement as simply rotation on an axis while standing still. The waist can move in time and space without rotating. As other posters have written, it can bend, tilt, open, close, and all while in rotation, btw, and all while being carried by the legs. As I said before, I feel it is the [command]center of all movement/s. I'm just explaining my reasoning, btw, not arguing against any other position.
"To me, this works if vertical bending movement of the waist counts as waist movement and if one does not pause with the leg cocked. This is indeed how I have seen kicks performed on the very few occasions I have seen them performed by people whose form is similar to the Yang's. Unfortunately, the Yangs seem to have no problem pausing after straightening their standing leg in a way that would seem to dissipate any energy gained by whipping the pelvis and the kicking leg under, forward, and up into position. Do you all see this differently?"
Well, I was also taught to pause, but we're talking about doing the form and there may be a difference. I was also taught, at first, to put my foot down at the end of the spin, sometimes with the explicit instruction to "balance yourself." Later, I was told not to put the foot down. Then, of course, the form was done faster, or the movement was isolated and done quickly. For me, it became more apparent why the foot was used more when the form was done more slowly. Maybe the same is true for "cocking the leg." Besides, "even if the movement is disconnected, the intention connects it."
"Since my hamstring flexibility is severely challenged and I have difficulty with being really sung in the kicking position, I cannot really decide whether all of this has much power, at least in relation to a Karate kick."
I odn't know how others feel about this, but I've never ime met someone who is a really good "kicker" who hasn't done them each (but at least one) literally thousands of times --and thousands of these against bags or opponents. That's another reason why I use the "walking" analogy. When your kicks become just "leg movements" --maybe better, "body movements"-- then, it's not likely you'll be able to kick effectively. There's no secret to a Karateka's or a TKD's kicking ability. OTOH, learning how to kick effectively is not necessarily the objective of all practitioners. And, of course, for those over 25, it's better to start slow. BTW, I've seen people stand up somewhat to do Sweep Lotus, but I think I agree with Nick that the supporting leg should remain bent.
"Many styles seem to use the springing action of the standing leg as an important component of the kick."
Well, true, in this case, if you straighten the supporting leg, you will lose some of its "spring" energy. Anyway, all this is just my .02.