I think the name of institution will only really matter as perhaps an initial boost, if that. By the time you are applying for an academic position, you'll have buffered a "dubious" MD/PhD program (there are none, of course) by a residency and/or fellowship. The medical school name becomes less important than the residency and fellowship and what sort of research you've done. 5-10 years into a tenure-track position I would think it's pretty irrelevant what medical school you went to. I think Harvard and perhaps a handful of other schools may be very choosy and have inbreeding tendencies, but most (even US News top 10 and top 20 schools) don't care.
I know a guy at a top 5 medical school/center who was offered the chance to become chairman of Harvard's pharmacology department (he isn't at Harvard). He's one of the foremost members in his field. He's known for what he's done and his 500+ publications. Nobody cares that he did his medical school and residency training in U Illinois rather than JHU.
On the other hand, I think where you do a PhD (if PhD only) matters MUCH more for getting a faculty position. If you look at the list of physics professors at Harvard, you'll see that they're nearly all Harvard, Berkeley, Stanford, CalTech, Chicago, etc. and a very small smattering of a few other places. It's highly elitist and inbred, and I think that's the case with most of academia.