So, as a former MSTP student I will chime in. The MSTP's organizational team (usually a program director, a couple assistant PDs, program coordinator, and secretary) are extremely important. As an MD/PhD students, you are caught in this nether-world where no one except your own program knows exactly what you are up to and your requirements are. These people are going to be the ones who help you navigate the day-to-day of your next 8 years. Key roles for these people include:
1) helping your program stay funded
Not much more explanation needed, but for your program to continue to exist the NIH requires a lot of paperwork.
2) recruiting future MSTP students who will ultimately make you look good or bad
An awesome PC can help your program recruit better students. Better students become better residents and ultimately faculty. These are the connections that will help your career.
3) making sure you get paid (in a timely fashion)
Paying MSTP students is a financial quagmire. Departmental funds, PI grants, medical school slush fund, etc. MSTP students are paid from a variety of sources during their 8 year career. And this is at programs that have NIH T32 grants! Just wait until that time that the end of the month passes and somehow that direct deposit never showed up. The difference between paying your mortgage on time and not is them getting the situation rectified.
4) coordinating your requirements through multiple departments/programs
Who figures out how much PhD credit you get for medical school? Or what happens when your PhD department changes your degree requirements while you are in medical school? These people can save you a lot of time by doing this stuff for you.
5) dealing with any unexpected problems which arise during your 8 years
This is a big one. For instance, my PI left the institution during my PhD years. My program arranged for me to keep the same PhD committee and project while changing advisors and staying at my current institution. This involved vetting a new PI at the institution and transferring a grant. My program handled the paperwork without a flaw.
My friend, on the other hand, also had a PI who left. He wanted to stay with his PI and project but get his degree from the current institution and come back for M3 and M4 years. The program made it happen.
#1-2 are indirectly important to your future, as they will help maintain the value of your degree. #3-5 are directly important and will affect your day-to-day. All this being said, most programs have good leadership at both the faculty and administrative levels. I wouldn't somehow rearrange your preference too much based on perceived organization, but a program that just can't manage to get it together can cause problems for you. Other than my 2 PhD mentors, the single person who had the biggest impact on my 8 years was the program coordinator. Keep that in mind.