I'm not buying that, especially since you've provided no sources.
E.g. Crain's article from 2016 says that city gun permits soared over the previous 5 years to... 62,000. That's in a city of
8 million. I.e. most of the firearms are coming from out of state, notably straw purchases in lax states like Virginia.
The article also points out that gun laws in New York State have become
more strict. The SAFE act, passed in 2013, banned new sales of assault weapons and large magazines; increased background checks; and stiffened penalties for gun crimes. Everything survived court challenges, except the part that made it illegal to put more than 7 rounds in a 10-round mag.
The harder it is to buy a pistol, the more New Yorkers want to get one | Crain's New York Business
NY Times last year put the number of permit holders closer to 42,000. A big chunk of that are security guards, or people keeping a firearm at a business. NYC still has a
17 page application for a handgun permit. You have to be interviewed by the NYPD; renew every 3 years; meet strict safe storage requirements, and so on.
https://www.nytimes.com/2017/10/04/nyregion/new-york-today-what-are-new-yorks-gun-laws.html
NY / NYC laws did not change in the wake of
Heller or
McDonald. E.g. in 2012, the courts upheld NY state laws that set strict conditions on concealed carry permits. I know of no NYC laws struck down in recent years. I see no indication of NYC having a major backlog on permits or renewals, nor am I aware of any case that required the NYPD to clear any such backlog.