Suburbs are more welcoming to kids in general because here we have more parks for kids than dogs. Here we don't have museums - but we don't miss them because we know from experience our kids can only stand being in one for about 10 minutes before they start doing that will get us kicked out if we don't leave first. A good portion of your art not kid friendly. Culture is everywhere, we don't get the "high culture" of the cities, but we have our own culture that works for us. We do a lot more cooking at home as we can't afford eating out as much, but I will grant when we do go out there is a lack of choices. All you really have that we would even use is much better transit.
It doesn't have to be that way. I'm told NYC is still welcoming to kids. Most cities though are not places where they have things for family.
As a kid, I loved going to the natural history museum and the science museum. There's plenty of museums that caters to kids and that are really great with kids (not sure in the US to be honest but at least that's true in Europe). Museums doesn't mean art only.
Likewise for theater plays, that's the kind of things that are only available in bigger cities but can be tons of fun with kids.
Living in the city is great; access to museums, art, culture, food, and transit is tops.