> the mechanics and the why are closely intertwined

But the "why and what" must always be explained first even if it is incomplete; since that is the problem we are trying to solve. With logic it is even more important since you can follow a proof from one step to the next but by the time you reach the conclusion you have lost the connecting thread to the starting point (for most general folks) i.e. "you have missed the forest for the trees". This is why many folks feel lost when doing mathematics as mere symbol-pushing.