I don't know about other SDNers, but I'm from a state school, and I did not find that it had any true bearing on my success in the application process. Even though I was at a (not great) state school and not even in the Honors College, I carved out a name for myself doing a lot of ECs like leadership, research, and volunteering (clinical and non-clinical). I also had a good GPA (3.8), but did so-so on the MCAT (28R). Overall, I got into 7 med schools, both state and private, low tier and high tier. I really don't think that your undergraduate institution confers any type of advantage/disadvantage. It is what you have done with your opportunities at that institution that makes the difference.
That's just my experience and opinion. Go where you feel comfortable about the education and experiences you will receive. And don't forget that many experiences will be outside of any particular institution anyway, like summer programs, and study abroad programs. You make of your experience in undergrad what you will. You dictate your experience, not the institution necessarily.
Good luck!