If you have to get a permit to exercise a right, that right is being restricted. There's no such thing as an "unrestricted permit"; the term is an oxymoron.
That said, I "carry" a lot more than you do, apparently. I just don't find it to be a big pain. YMMV. Obviously.
Most NY permits are restricted to “Target and Hunting” or “Present Employment” or “Premises”. It sucks, but it is what it is.
Hence “Unrestricted” may mean something different here.