Yeah, setting this in NYC in 2001 means you will have a big, 9/11-shaped elephant in the room that people are going to be constantly thinking about. Perhaps they may even expect 9/11 to be a plot point in your book. If you don't plan on including 9/11 at all, or being set in a post-9/11 NYC (where things are going to be VERY different), then this is going to be an expectation pretty much every (American) reader is going to have of your work and it will really alter how people process it.
If 9/11 has nothing to do with it, couldn't you do 2000 instead?
Regardless, "stop and frisk" was happening at the time, so there was a lot of racial profiling going on (I know that sorta implies that there is no longer racial profiling going on, but there still is, because acab, but stop and frisk was a uniquely horrible way to do racial profiling). A lot of resources wasted on terrifying innocent black and brown people.
Also: NYC has ridiculous gun laws. It currently has a very low murder rate because of that, so you'll have to look into which laws kicked in at which times.
Who got taken out by the hitman? A pretty young white girl? The son of an investor? A sex worker? A """""""thug""""""? The police ALWAYS have varying degrees of giving a shit depending on the "value" of the person who was assaulted/killed, so, y'know, keep that in mind, too.
Also keep in mind that whatever you're seeing on Law & Order is copaganda and in no way reflects the actual attitudes and procedures (and case closure rates) of real PDs/DAs. Unless you want to also make a piece of copaganda, then probably ignore everything I just said and make whatever it is you want! As long as you make the cops the good guys and "the bad ones" are few and far between (and always get brought to justice/disciplined properly), then they (and, sadly, many Americans) won't find a problem with your story.
Fun fact: Saw 1 (diegetically) happened on 9/10/2001, so, uh, that's probably why no one really went after Jigsaw!