Having read the document, it is not totally accurate. It states:
Some nuances. You need to get a permit to even possess a handgun in NYS (not discussed NYC). You have to justify getting one - unrestricted, self-defense, competition, hunting, premises. Counties vary on their strictness. One might have challenged on the basis of needing justification for any purchase, not just carry. So I don't know why they went this way. It seems the need to have justification for any gun is a stronger Heller case but they also want to free up carry - so maybe that's it.
Now, the virtually impossible statement is not true. In Western NY, there are thousands of people with unrestricted permits. Other counties in the state are similar. You have to take a course, get references and justify but you don't need a detail justification (I'm a diamond salesperson as in NYC). Wanting to defend yourself and stating it reasonably along with competing, hunting, having some knowledge will do it. I took the class with about 50 folks and the outfit that gives the class is very successful in the students getting permits. The local Cabelas gives classes - if it were impossible that wouldn't be the case.
I agree in theory that justification is unconstitutional (but what is is decided by Scotus),. however, there is large handgun culture if you get away from NYC.