As someone who's recently been through the residency selection process, please do not listen to these people telling you that the tier of your medical school is less important than most other factors. The ranking of the school you go to plays an enormous role.
This works out well for you, since you've somehow gotten the (IMHO erroneous) idea that NYU is more highly regarded. Sinai's internal medicine residency program is probably considered a full tier higher than NYU's. Going to Mt Sinai over NYU would very likely make it considerably easier for you to get into many higher tier residency programs.
I'm no stranger to the program directors' supposed rankings of applicant characteristics, but I can tell you that many of them are just plain misleading. People tend to have similar MSPEs across the board, save for the last "money" line (outstanding recommendation vs excellent, great, good, etc.), and people putting forth an average med school performance tend to attend a residency program within the same tier as their medical school. Those who move upward or downward tend to have done something remarkable, in either a good or bad way.