I agree there are many problems in the approaches done in many of the calculus books used but I disagree about the mean value theorem (Lagrange theorem for me). It is the cornerstone of analysis. You probably have some treatment in mind or a whole list of them. 
Lagrange theorem have the combined power of Bolzano's theorem (continuity of the reals, for what is worth) and the notion of derivative. If you want to pass global info from the derivative to the function, the mean value theorem is the place to go. Of course one has to be clear that the problem is really about the "how" are things presented. After all MVT is equivalent to the continuity of the reals. 
One trivial change that I always try to do is a simple change in the writing. Instead of writing the equation with the derivative isolated in one side write the function isolated (like a Taylor). Also with the definition of derivative. instead of writing the derivative in one side of the equation writing it inside the limit. That apparently unimportant change has as outcome that students grasp better the connection between them: generalize MVT to Taylor, derivative to differential, use MVT in application.