Grothendieck is/was (according to Wikipedia, he is still alive, but he retired from mathematics years ago (http://en.wikipedia.org/wiki/Alexander_Grothendieck#Politics...) a great mathematician who knows/knew a lot about prime numbers, but he could not apply that 'to every problem he came across'. In fact, when pressed to give a concrete prime number, he came up with 57.
That makes him a counterexample of the claim that all great mathematicians can apply heir knowledge.
If he were a computer scientist, he would have written several brilliant papers about the pros and cons of various properties of programming languages, but he would not be able to write a compiler for any language or even to write a program.
I'm confused. What is your basis for that statement? If you you know how they think why don't you just express it? :-S I am sorry, I have no clue what your point is, or even if you have one.