Hm...here are my thoughts:
1) There are a number of reasons why someone would necessarily want to go to a top tier school and it's not just for the brand name. For example, people who want to go into academic medicine or pursue the more competitive specialties(radiology, opthamology, anesthesiology, dermatology) definitely benefit from going to a top tier school. Does going to a less than top tier school necessarily preclude you from pursuing these options? No. But just like with med school applications - if you come from a top tier undergrad, you may get an interview just by not having any glaring FLAWS on your application; whereas, if you come from anything less, you may need something glaringly EXCEPTIONAL.
2) Patients can see where doctors go to medical school/do their residency and some do care. Does that necesarily mean those who go to the best schools are the best doctors? Again, no. But when you know nothing else about a doctor and are choosing blindly, many patients will choose based on what they know - name brand schools.
With that said, do I think it's a good idea to reapply? It really depends (on where you applied, where you got in, etc. etc.). Up until recently, I was strongly considering reapplication (even though I had acceptances) also. I was dealing with a lot of personal things, applied late, didn't write the best application, and made pretty arbitrary choices when compiling my school list, and only applied to a handful of schools. However, if I did reapply, I'd come back with a stronger (even if you're top tier material, you can still improve) application and apply from a different angle (ex. applying m.d./ph.d, applying to schools that have less/more of a research focus, or maybe applying to schools that have a strong program in caring for the underserved, etc. etc.). I absolutely agree that "I could've done better" probably isn't a good enough explanation.
Keep your fingers crossed though and don't give up! If you're on that many waitlists, writing LOIs may help. And don't worry, even if you're a horrible interviewer, different schools weigh the interviews differently. I.e., I'm pretty sure UPenn interviews do not hold that much weight. UPitt...I'm pretty sure wholely disregards the interviews (ok... that may be an overstatment). Cleveland Clinic seems to put a much heavier emphasis on the essays than the interviews. and Baylor seems to HEAVILY emphasize interviews (unfortunately...).
In anticipation of any backlash - these are just my thoughts and are mostly based on anecdotal evidence. If I made any statements without a qualifier... mentally insert one in.