Thought does not move the body.
Yell at your hand and demand that it move! Does it? Of course not... Yet, assuming you do not lack certain ordinary functionality, you can and do move your hand at will, without thought and without concentration. What is it exactly that does this?
And what can possibly interfere with it?
What sensory perception is your left foot providing you right now? Without calling your attention to your foot, you may not have even consciously been aware that it was there; but, now that it is, what do you feel? How about your right hand? What does it feel? And when you brought your attention to your hand, what became of your foot? Bring your attention to your your left hand. What does it feel? What is 9 x 6? When you did the math, what happened to your hand?
In the middle of the night, walking on a dimly lit path, a thin wavy shadow on the ground catches your attention. Seeing a snake, you momentarily seize. Realizing it is only a stick, you gather your composure and continue on. What is it that made a snake out of a stick, and what made the body respond as it did?
In your own martial arts practice, are you simply learning rote mechanical skills, or are you also challenged to consider things a bit more deeply? Can you imagine the difference such practice would have on your training?
What might "integrating mind and body" really mean in practice?