Brilliant title...love it.
Straight away I can see there will be 'versions' and misunderstandings on what you meant by thinking. I agree, that mind or aspect of mind has to dissappear for correct mind and body co ordination.
Concentrated thought if you will is not the same thing as thinking.
Perspective is that in all sports the optimum is mind and body co-ordinaton.
I must admit that over the years there have been those from very physical perspective and those from a Ki oriented perspective, no problem teaching wise. Ahhh, but then we come to the great theorist, the thinker, the over intellectual.....problem.
So I have found the 'thinker' to have more to learn and less open to letting go and the discipline necessary overall.