A general inefficiency example using two airpots I often have in mind: IAH and JFK. In IAH, traffic is spaced out in cruise well before the arrival begins, where they're burning much less fuel at high altitude. They're then instructed to descend via the full STAR (standard terminal arrival), including the transition to the approach for the runway in use. This looks like a hundreds mile long line of planes all following the same path to the same runway. The only further instruction necessary from ATC is speed adjustment, and allows for an almost continuous descent to the runway provided there's no weather in the way.
At JFK, the spacing occurs much later and, even though the airport has some similar arrivals (though none quite as thorough), they instruct every altitude change and speed adjustment for every aircraft individually, along with vectoring them to the approach for the runway in use. It results in an incredibly busy radio environment and numerous inputs from pilots flying planes capable of simply following a STAR and transition, like in IAH (or ORD, SFO, LAX, etc). There's also much more time spent at low altitude burning more fuel. It reminds me of everything else in NYC, which seems to be inefficient on purpose (NYPD, MTA, DSNY, etc.).
JFK and IAH have a similar number of aircraft movements. Yes, NYC has busier airspace with numerous airports, however, I don't see how this prevents the design and implementation of more modern procedures. Flying around at 4000 feet in an airliner getting numerous vectors just seems ridiculous compared to what I experience elsewhere.
This whole comment seems like a charming case of advanced Dunning-Kruger. You sound very confident in your assessment, but then it seems you are honestly suggesting that NYC TRACON controllers are being inefficient on purpose… for what? Fun times? You said it yourself— it is a busier airspace. The approach and departure spacing of EWR, JFK, LGA and TEB— not to mention the changeable airspace restrictions over parts of NYC itself and managing GA traffic into and out of the Hudson River Corridor— makes a simple single hundred mile long elephant walk to your preferred runway at JFK logistically untenable. Not to mention potential conflicting traffic from smaller airports in Connecticut and Long Island… And just a gentle reminder that as pilot in that airspace you have precisely one small piece of the big picture when it comes to being aware of exactly how complicated that can be for the ATC personal.
This seems like the kind of critical infrastructure whose uptime requirements can make upgrading challenging once in place.