After just a few years into ownership, mortgage becomes cheaper than renting so it is the homeowner who will have extra cash to invest for the rest of their life, not the renter.
Of course renting will be cheaper than buying an equivalent place. Most of the time in most places that's the case.
My point was this is true in the year of purchase and for a few years later. Over time (and you'll need to live somewhere your whole life) it will become cheaper. And then a lot cheaper. Rent will only ever go up.
Also, if you're buying in NYC you WILL get recurring and increasing monthly costs.
That $800/month HOA? Yeah, it's not gonna be $800 in 5 years.
Buying is not the same everywhere.