MT classes teach and practice the spin recover after a missed kick. Thats the big diff between a karate kick and a MT kick. The karate kick is done with the leg powering the kick and MT is done with the rotation of the front leg, hips and torso to whip the leg. This may seem inefficient, however the MT kick is potentially devastating and when done correctly no can defend.
Bold 1: Actually the biggest difference is the set up. With the "Karate Style" the front foot rotates at the same time the torso/pelvis. While with the "Muay Thai Style" The foot is rotated in advance, that is where the added power comes from. When the foot is able to plant in advance, there can be a more predominant weight shift behind the kick instead of as part of the entire motion. And its not inefficient at all, as you implied, its actually a more efficent way to transfer energy, however its slower and because of the additional weight in the kick, and in some cases the body is forced to spin to bleed of the excess the leg muscles cannot counter. These are cases where this style is flawed.
Bold 2: That's not Muay Thai, its basic roundhouse kick form.
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