NYS PL400
6. License: validity. Any license issued pursuant to this section
shall be valid notwithstanding the provisions of any local law or
ordinance.
Notwithstanding = Despite.
So a local jurisdiction can make up any restricted places they want, but they don't matter. Your license is good anywhere in NYS, except NYC and schools, as laid out in...
S 265.01 Criminal possession of a weapon in the fourth degree
(3) He knowingly has in his possession a rifle, shotgun or firearm in
or upon a building or grounds, used for educational purposes, of any
school, college or university, except the forestry lands, wherever
located, owned and maintained by the State University of New York
college of environmental science and forestry, without the written
authorization of such educational institution;
and the usual federal no-nos (courts, federal buildings etc.)