I have just been teaching calculus for the first time, and I am firmly of the opinion that in many calculus courses, the mean value theorem should have essentially no role. All the applications of it can be explained intuitively without any reference to it, and the semblance of "rigor" that using it provides is largely obscured by the fact that it is presented as merely a black box without a proof of its own. It's a fairly arbitrary principle from which to rigorously derive basic facts of calculus, and if we were to provide completely rigorous proofs of everything, there are other, more intuitive proofs that do not use MVT (typically using instead a compactness argument, the technicalities of which can easily be brushed aside while leaving the key idea of the proof clear).

That said, this impression is based on my one experience with a particular course which talked about MVT but only quite briefly. I could imagine it fitting well into a different course which was overall much more rigorous and which went through Rolle's Theorem and MVT in greater depth.