I am going to echo much of what previous posters have said. I am currently a Ph.D. candidate in a non-Ivy but still well-ranked Political Science program. I did not do an MA, but several people in my cohort did. The only possible justification for getting an MA in Political Science is to better your chances of getting into a good Political Science Ph.D. program. Even then.... there are other ways to make yourself a stronger applicant for the Ph.D. that do not involve handing over a ridiculous amount of money to get a degree that has basically zero practical application on its own. A Poli Sci MA will not help you get any sort of job outside of academia. It *might* make your law school application stronger, but the law admissions committee will consider that MA degree alongside a host of other, potentially more consequential factors and life experiences when evaluating your application against others.
As a previous poster said, MA programs in Poli Sci largely exist as cash cows to fund Ph.D. students. If you have the money to burn, go for it. If you have to take out any sort of loans, I would strongly advise against it. Because you brought up "access to faculty" specifically (which is a very legitimate concern), I'll just say this- in a place like Columbia with funded Ph.D. students and "academia famous" faculty, you will never be high on any professor's priority list as an MA student.
I know nothing about the MAPSS program at Chicago. But I cannot imagine a way it could possibly be worse for you than doing an MA in Poli Sci (regardless of the prestige of the department), given that you intend to go to law school, not do a Ph.D.