1. I don't deny this point, and I admit that I thought about this when I first made the decision to come here (as opposed to Brown and Cornell, for example). You're bound to get that with any med school that isn't attached to an university. Lay people have name recognition of these places largely because of the university, rather than the hospital itself.
2. So it's really up to you. If it matters to you that your barber or your great-aunt knows of the medical school you go to (or for your parents to brag about where you went to school with their colleagues), then this is a legitimate factor for you. If you can trade some of the "glitter" for the lifestyle and the experience, then hey, why does it matter?
3. Mount Sinai is surprisingly not as obscure as I initially thought it was going to be. I'm from Asia and when I go home even people in Asia (who don't work in medicine) have heard of Mount Sinai. If you're north of DC, I think most people know on the east coast, as well as the major cities in the west coast. The hospital does pretty good PR to get its name out there, IMO. Maybe we'll be like the next Mayo in 10 years 😉
Honestly, for me personally I was a bit sad in the beginning when I wasn't going to a "famous" school, but then the Sinai environment and the ability to go to school in NYC quickly outweighed that. Finally, I went to a top 5 undergrad, so I felt like if I needed to "show off," I got my share of that in college, so it's not a biggie 😛 (everyone's shallow/vain to a certain extent, it's ok)
Oh and since Cornell's revisit falls on the same weekend and for the people thinking of picking and choosing - I was on the revisit committee last year and I looked at this year's schedule (same as last year's). If you want to split up the weekend, I highly suggest sticking through all of Friday's programming (that's where the "fun" is), and staying till lunch on Saturday. I thought Saturday afternoon's activities are more "meh," and since Prof. Laitman is speaking on Saturday morning you really don't want to miss that, and also there's an admin panel or something like that so you can get all your questions about curriculum/housing/research/finaid and stuff answered, which should be helpful.