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. 

--more added after the first comment---

Oh, that's true. Nevertheless the two are related. Of course the question about teaching is not a well stated one. It depends on the goals of the course. What is it that you want your students to be able to do. One unavoidable one in a calculus course is to study functions, nice functions. Nowadays, this nice functions tend to be continuous and differentiable functions (although a close look at most the courses tell us that the class of interest is much narrower.)
The concept of derivative seems to be then, required. Although I bet a good course can also be planned with the notion of power series instead (which seems to be returning to times before Newton-Leibntz but i am not so sure [ask Doron Zeilberger about it]). 
A less chocking approach is to put both concepts side by side. And MVT is a way of linking them link. 
I have to say that the way programs evolve is by taking the old ones and doing little "improvements". It is true that in the scope of basic ordinary calculus course you can skip MVT only loosing the possibility of asking problems like "prove that $|sin(x)|\leq|x|$". 
But again, it is a problem of goals. It is also good to take into account that learning process works starting from the horizon of already acquired knowledge. Even is students at the end they not even remember the statement, not to say how to use it, It prepares the ground for further development. Or just remember, you the working mathematician, how many times (if no every time), you have gone to a conference in which you don't understand a thing, in which you only remember two or three names at the end. Now remember how many times just knowing that that name (or word) exists or that is related to some other name has opened a complete new road in your research. It works exactly the same for students, even though they are not doing research. I say this to point out that the validity of an element in a curriculum program should not only be judged by the ability of students to actually get it but also by the grounds it creates to build on top of it. Education is a process that involve not only teaching but also evaluating. Maybe it is better to look at how MVT is evaluated instead. If it is playing a role of a connective element then it is wrong to evaluate the skills of applying it (which involve both understanding and skill). Changing the evaluation method is a less destructive approach than elimination from the program. 
Uff, I have written to much. If I forgot to say something I will say it later.