This is coming from a current applicant, the least qualified opinion, but:
During my post-bacc years working in research, a physician in our lab invited me to join him in the clinic. I became a regular visitor, and strongly feel that the experience was really important in understanding the clinical significance of our projects and gaining clearer insight into the translational prospects of our work. So in that regard, I can see why a student in an MD/PhD program might seek out such experiences. I see where you are coming from. Related to Neuronix and Maebea's points, the student would have to make sure that this time is spent learning about the clinical realities that are relevant to their chosen thesis and course of study.
However, I don't see any reason for this activity to be formally by the institution. You would probably be able to set this up on your own regardless of where you are studying. More important would be to choose a place with an academic environment that has strengths that align with your interest. (I would expect schools usually have clinical programs that are consistent with their research strengths, but I may be wrong on that?)