Edit for OP: The difference in prestige between the colleges you mentioned is small. You should decide based on other factors. The remainder of this post is relevant for the generalized question "does undergrad location matter".
Undergrad institution definitely matters. A Harvard Med student on the admissions committee showed me a breakdown of undergrad institutions for the entering classes of a few years (2010 through 2013, iirc). Harvard has 165 slots total (New Pathways plus HST). Each year, about 10 students each come from the big names: Princeton, Columbia, MIT, and Stanford. More than I can count come from Harvard College. Cornell may be an Ivy League, but it fares much worse. Only about half the class comes from "other" undergrad schools. Boston University seemed over-represented, possibly because BU undergrads often do research under Harvard Med professors at Boston hospitals. Getting that LOR from a Harvard professor probably helps.
That said, roughly 90% of the applicants from Columbia, Princeton, MIT, and Stanford get rejected from Harvard Med. So undergrad location matters, but everything else matters, too.
Is this discrimination? 10% acceptance rate for top schools vs. 1% for other schools? Maybe not, the Princeton and Harvard grads may have much higher MCAT scores and much better ECs. How do you think they got into Princeton and Harvard for undergrad?
Having a Harvard diploma will not make up for a bad MCAT score. And Harvard GPAs are actually somewhat inflated relative to most US colleges. I believe the most common grade at Harvard College is A, and the median is A-. More than half of Harvard College graduates have honors of some kind (cum laude or higher).
If you turned down Harvard College and went to University of Home State to save money, you definitely have a shot at a top-tier med school. Going to a good undergrad school is just a proxy measure. Correlation does not imply causation.
