Maybe I'm misunderstanding you, but your statement appears to be incorrect to me. "Big movement" training is not necessary to make "small movement" work correctly. Boxing and Wing Chun do not use "big movement", and work very well. Even in grappling terms, things like Wally Jay's (small circle) style work as well or better (IMHO).
Can you clarify?
I guess maybe the better way to state is is that according to the theory and foundation upon which these systems are built, Choy Lay Fut and Tibetan White Crane as good examples, this is how it works. The power is based on a very large pivot and rotation of the body, sometimes twisting of the torso tho I don't want to say that too strongly because it might imply an injury-prone movement and it is not that. At any rate, power is generatedy by rotating the body and building a huge amount of striking potential at the end of the arc, in the fist. The big movements really capitalize on this, and it engrains the type of delivery into the muscle memory. In some ways, it can be seen as being exaggerated, but like I said, it's a training tool and not necessarily how you would want to deliver it for real. Altho given the right circumstances you could really land some heavy blows with it. At any rate, once your body had really internalized this method, developed thru the use of these big movements, you can then shorten the movement while still getting the same kind of power. This works because the movement is initiated with the feet bracing against the ground and driving up thru the torso, not in the hands that are swinging out there. Understanding how to initiate the torso removes the necessity of using the big movements.
I don't know much about boxing, but I do have some experience with wing chun. The development of power is different, not relying on the same kind of pivot and rotation. My experience with wing chun is that power is developed largely by squeezing it out thru short, explosive movements, tho the training of this power can be slow, drawn out movements. In a way, it's kind of the same thing, but with a different concept and origin of where the power comes from. Long, drawn out movements (as practiced in Siu Nim Tao) turn into fast, explosive, powerful short range punches.
Now, getting back to the pivot and rotation that I was describing above, I know that many systems do this also, and I won't deny it. It's a foundational part of things like the Reverse Punch found in many systems. But, from my experience, systems like Tibetan White Crane and (as far as I can tell) Choy Lay Fut, make this a heavier focus and use a more "exaggerated" movement than I've seen anywhere else, in order to develop this. In White Crane, which I study, we focus on that pivot and rotation in just about every single movement that we do. The first thing that a beginner would learn is the pivot. Just stand there and get the pivot right, the whole body moving in unison, beginning at the feet, to drive the pivot. Later, a more advanced and probably more useful version of the pivot is learned where the feet root into the ground to drive the pivot which manifests primarily in the hips and waist, paying particular attention to the stances to be properly grounded for driving the power.
So, we use big movements to develop the technique, but once you are skilled with that, your actual delivery can be with a much smaller movement.
I'm not going to say that it's better than other methods, but it works well for us and we do deliver tremendously powerful, even frighteningly powerful, strikes. Once I began working these strikes on a heavy bag, I felt they were significantly stronger than before I began studying this method. Maybe my prior instruction was deficient, I dunno. But I have seen a real difference once I began to learn White Crane, that's just my own personal experience.
So, what about other systems that use small movements, other that what I discussed above? Can't really comment too much about them since I don't know much. I don't know have any experience with Small Circle JJ, for example. I'm familiar with how we use the pivot from my Tracy Kenpo background, and we actually have a few things in there that have some similarity with the Crane methods. I personally feel that those items ought to get a heavier load of the attention in order to really develop their potential. In some ways, this is connected to our earlier discussion of basics, and stances. Taking the time to get the stances just right, and the timing and body-connection with the pivots, can make a huge difference. My Crane training gives me a bit of a different perspective than most kenpo folks, so I see things a bit differently, but there ya go.
My comment about Jamey's comments was just that, the big movements in arts like Choy Lay Fut are there for a reason, and it's important to understand what the reason is, before making a judgement about them.
This is something that can be difficult to describe in writing. It really needs to be seen and felt in order to understand it. Hope I managed to get something across. I'll leave it at that for now, otherwise I might end up getting a bit long-winded and it might not really clear anything up. If you've got some more questions on it, I'll try and clarify some points.