I think that most powerlifters do hip belt squats if they feel that they need to bring up their quad strength.
I have never felt that doing leg extensions of any sort would help a squat. That is similar to saying that a tric kick back will help to bring up your flat bench. It might....but you could do other exercises that would do it more.
The squat uses so many different muscles just to do the exercise properly. If I am looking for an exercise to increase my squat and I have to choose between the leg extension or any other type of squat...I pick the other exercise everytime because it will have the greatest carry over. The quads go through a different type of movement in a squat than they do in the leg extension...it does not serve the functional purpose needed to improve the squat.
I think that more people fail on a squat because of their hips, hams, and back strength...not because of quad strength. Why spend time training the least limiting factor?
Lots of people focus on exercises that are for the muscles that they can see. The main focus of a lot of people are the shoulders, bic, chest, abs, and quads. The strength comes from the tric, rear delts, back, erectors, glutes, hams, and calves...those are the limiting factors...so focus on them.
B True