Don’t you have any idea of what decile that 3.7 UW GPA puts you at Stuyvesant? Certainly, Stuyvesant must send colleges a nomogram that gives them some idea of where in your class that 3.7 places you. And I’m sure that colleges understand that since Stuyvesant is admission by competitive exam, and that it is the one that requires the highest score of all the NYC exam schools, a 3.7 UW and a 1560 SAT correlate, are not an indicator of an underperformer. After all, the student body there consists only of students who were high achievers before they began Stuyvesant.
Honestly, I think you’re not overshooting in terms of selectivity in your college list. Schedule a meeting ASAP with your college counselor to discuss your chances coming from Stuyvesant, based upon recent years of acceptances.