This is my opinion about the "prestige" factor. Let me know if you disagree considerably with any particular point.
Some of this prestige is important for bragging to your friends, but that's about it. My parents have no idea what Cloister or HHMI is (they're not in the medical academic circle), so they would likely be more impressed by Harvard or Yale because they recognize the name. But for those people who will be making decisions about my future residency, etc., they will undoubtedly know that all these programs are great places to do research and will likely assume they are equally competitive (I know that is a loaded statement, sorry). But regardless of competitiveness, I hope the most value will go into the quality of my research and my productivity, which is not necessarily associated with the particular program or institution, but instead the mentor and research topic. Therefore the prestige factor is most likely going to depend on who you're doing research with and on what topic, not where and through what program.
So I guess the point is who are you trying to impress with this prestige?