There seem to be several aspects to your question, I will try to address them as I understand what you are asking.
You seem to wonder about why the condition is differentiability on an open interval and not a closed one. Well, if a function is differentiable on a closed interval then certainly it is differentiable on an open one. Now, if a theorem can be proved from a weak condition then there is no need to strengthen the condition. In general one wants to impose the weakest possible conditions for a result. This is done so that the theory applicable to more situations.
The motivation for the creation/discovery of these and similar theorems from calculus is mainly to understand the relationship between a function and its derivative. In the initial stages of development of calculus there have been many subtleties that needed to be worked out and it took quite some time before a rigorous approach was established (by Cauchy and Weierstrass).
There is a lot of geometric intuition related to the study of functions. This intuition is sometimes helpful, guiding to correct theorems, and sometimes wrong, as witnessed by that or other unintuitive counter example (a famous case is the existence of a nowhere differentiable continuous function (first constructed by Bolzano (unpublished) and most famously by Weierstrass).
Historically, a lot of research and debate went into clarifying these issues and fine-tuning both the theory and ones geometric intuition. This is done by takings ones intuition, formulating a conjecture and then trying to prove it. The mean value theorem is very intuitive geometrically and indeed can be proved.
If you can get your hands on the book Adventures in Formalism by Criag Smorynski, I believe you fill find very interesting examples there from calculus that may help you understand how things work.