You might say "aha! walking is not thinking", but surely you would think very hard about where to put your limbs if you were to walk through a minefield, etc. Is walking thinking if the conscious mind does it? So whether "thought" can occur unconsciously or not is a question of English not of neuroscience.
I think this statement makes the same error as the article, just in reverse: making a very strong claim that's impossible to prove.
I don't think walking uses "thinking". That sounds incredibly inefficient. Wouldn't unconscious thinking be slow, just as conscious thinking? If not, why not? Wouldn't the body be better off using its short-circuited system, as opposed to thinking through it?
And consider animals whose brains are not very large or known for their mental ability, yet who are very good at body kinematics. I'm inclined to think these are not related at all. The body likely uses some short-circuited system for kinematics, similar to how spiders' legs work, just more advanced. If we figured out how such a system works, we'd be able to build very good robots, but right now we try to make our robots think about how to control each individual part of their limb, so it's cludgey, slow, and inaccurate.
> but surely you would think very hard about where to put your limbs if you were to walk through a minefield, etc.
I'm not thinking of how to put my limbs there, only where to put them, which is a very different task from actually putting them there.