What are "the suburbs" here? Westchester? Nassau? Which parts of Long Island? Which parts of New Jersey? Each has different characteristics. Likewise, NYC property taxes have characteristics all of their own.
NJ is known as a high property tax state. That's true but you need to understand why. Why is because so many NJ residents work in NYC and pay NYS income taxes rather than NJS income taxes.
As for the high-tax NY suburbs like Westchester... it's complicated but the tl;dr version is school spending, inefficiencies and state-mandated defined benefit pension plans [1].
As a general rule though, NY state single family home ("SFH") owners pay disproportionately low property tax rates compared to those in multi-dwelling units ("MDUs") ie apartments, which is NY state's (fortunately tamer) equivalent of CA's Prop 13. Upstate SFH dwellers have voted themselves tax breaks in the form of caps on property taxes. Apartment dwellers however have to property taxes based on Assessed Rental Value ("ARV").
A $3m brownstone in Prospect Park might have property taxes of ~$700/month. A $3m condo in Midtown Manhattan will likely pay $1300-2000/month in property taxes (so ignoring common charges). And that's AFTER the condo and coop tax abatement, which is a "temporary" decrease that's been continually passed by the NYS legislature every few years continually since 1996. Without that, those apartment rates would be up to 50% higher.
But it gets worse. Those who pay the lowest taxes of all (as a portion of their property values) are... foreign billionaires who buy ultra-luxury condos on 57th Street, CPW, Park Avenue and 5th Avenue.
I applaud your effort to correct the GP's misconceptions but you've also made some sweeping generalizations yourself.
[1] http://www.westchestermagazine.com/Westchester-Magazine/June...
Some other forms of tax are much higher in the city, for example income taxes on higher earners, and some corporate taxes in the case of NYC, but there are also an incredible array of services provided.
Your premise that city taxes are "always higher" isn't something you can assert without offering evidence. The reality is that taxes in dense cities are usually different than in less dense areas, and reflect different priorities. Higher or lower depends a lot on who you are and what you're measuring.