I think that one reason there's probably no clear answer on this is that it varies from committee to committee, and within that, from individual to individual. Through talking with several members of various committees and admissions offices, as well as Professors with experience on admissions, I've heard that both cases are true.
My PI, a Professor I met at UCSF, and a former Dean at another school said that they WILL take into account that I was "from UCLA" (i.e. a highly competitive public school). I went through my CV with a director of a phD program at UCSF, and one of the first things she said was, "oh, you're from UCLA." It was a casual observation. I also saw a listing published in a Korean paper several years ago that outlined the number of students matriculating at Stanford and JHU from various undergrad institutions - the mostly highly represented were the top private schools and Berkeley and UCLA. Altogether however, those students probably made up half the class and the rest were 1 and 2 students each from smaller schools. However, other members of the same committees they worked on have told me about the process for grading and evaluating secondaries and it didn't sound like there was anything about school name calculated into their scoring formula for sending secondaries.
So, the short answer is that the answer is not black and white. From what I have seen from talking to people with admissions experience, for some people on some committees, being from a great school can give you a boost, though it is clearly not necessary to be from a "top institution" to get in, some people care some don't, there is probably varying opinions within committees. My personal opinion is the cream (no matter where it's from) will always rise to the top because schools will give you a holistic consideration. Then again, in this game, it never hurts to hvae every little leg up, but it's hard to tell how decisive your UG institution will be (it probably won't be). Committees will pay MUCH more attention to other aspects of your app and who you really are as a doctor. The point is to make the most of what you have wherever you are, do all you can do. From talking with committee members, the one thing that they emphasized over and over was they are looking for people with passion, drive, who will be leaders, etc (the stereotypical stuff) - if you just show them you'll be a great doctor, there's nothing else you need to prove?