I get a kick out of the original post.
First, the OP who has likely never wore a badge or otherwise performed any kind of police work is critical of the police
Second, assuming for a moment that there was an officer nearby and you hear a gunshot and see somebody go down would you prefer that they chase the gunman or provide first aid to the fallen person. Unless the officer was right there and saw it happen by the time he got to the site of the incident the perp was long gone.
Third, I'm still curious why anyone is concluding that it was an illegal gun. Perhaps I missed something in the original tweet but I don't recall seeing that. That being said even if it was illegally owned gun odds are they don't have a carry permit for New York City and hence that aspect is illegal. But then again that shows that gun laws aren't going to solve this problem.