I actually overheard some people in my lab today talking about NYU, I think one of the guys got wait listed and I heard the other guy say that NYU wait listed a lot more applicants this year because their US news rank skyrocketed over the past two years. NYU, at my school at least, is viewed as sort of a second class top medical school, as in its in the top 30 and is prestigious but is still inferior to the research powerhouses in the top 10. So they attract top medical school candidates but many of those candidates will get into other higher ranked schools as well and attend elsewhere. But if their rank went up significantly, then they might get more people choosing to attend which is why they wait listed so many people so that they don't accept more people than they have seats.
This is just what I heard though, but you might already know this since you've been so invested. But IMO, a well written LOI can't hurt, the worse that could happen is they not take it into consideration, so I would just send one.