Growing up, I had several of those in the suburbs in the suburbs too. And even then, my parents would take me into the city on weekends to visit the zoo and museums (I don't remember going to the aquarium, but we had one too).
But then again, where I grew up is very different from NY. My parents are both from NYC, and my cousins are both raising kids in the NYC area (one in the city, one in the suburbs), but my parents moved to Dallas a few years before I was born, and I grew up in Dallas but still got to visit my family in NY. NYC's strength is also the most aggravating thing about it: everything is in the city. This means that the suburbs are just bedroom communities with nothing to do. Visiting my family in Westchester has always felt like spending time out in the country. Dallas, on the other hand, is distributed and spread out. Except for a teeny tiny urban core, we're all suburb. We have edge cities everywhere instead of concentrating everything on one place. The suburbs where I grew up are also much denser than the suburbs where my cousins grew up, which helps.
One interesting study though is with my cousins in NY. They have three kids each, and both are fairly wealthy (honestly, they make more in a year than I'll ever see in my life). One of them lives in a swanky apartment on the Upper West Side, in walking distance of both Central Park and Lincoln Center. The other lives in a large house in Westchester, not far from where she grew up. I can say that, for my cousin who lives in the city, her kids have access to everything you mentioned, and she and the kids are very happy with where they live. But she's also filthy rich and is able to afford a very large apartment in a prime location. Not many people in Manhattan have that kind of privilege. Before she moved to the suburbs, my other cousin used to live in a teeny tiny two-bedroom apartment in Battery Park City. When she had twins, there was pretty much no space for anything, and as soon as she got pregnant with her third kid, she and her husband immediately began looking for a house in the suburbs, and they moved right after she gave birth. It's a giant house with six bedrooms, an expansive basement, and a backyard that won't quit. She told me that while she misses her short commute, it's all worth it to give the kids a perfect environment to grow up in. And the kids certainly have no lack of anything to do: when I visited last year, the kids were all involved in multiple sports leagues and other activities. Everyone is happy with where they live.