To me, the planning and density issues everyone's taking about feel like a red herring. Plunk this woman down in the middle of NYC and remove her need for a car whatsoever. Is she really going to let her kid go wherever he wants outside? Is she really going to feel a sense of community, or will she be like most New Yorkers and barely know who her neighbors are? (And before you say 'that's because NYC isn't planned well, either,' would these problems truly go away with more parks?)
I'm just theorizing, of course, but I suspect her real issues have a lot more to do with cultural homogeneity and societal mobility. A community with people from all over the place, constantly moving in and out? Low trust and lots of alienation. My whitebread little Texas town, where every church is full on Sunday and the average person's been here for decades? We're completely car dependent out here, the streets are real wide, and for that matter, there's no sidewalks and barely any parks - but there's unsupervised kids running around all over the place, and I can't get through a supermarket trip without being stopped for a conversation with someone I've met. It's nice, and as far as I can tell, it's all without the benefits of urban planning.
There used to be these things called small towns. 
One wasn't forced to choose between "20 minutes from anything" and "New York City".
Most people talking about the density issue are lamenting the lack of these middle choices. The places where a family would probably still need a car, but it would be conceivable to have a functional bus system to get you from one small town to the next, or into the city. Where you can still own a house with a yard, but walking to corner store wasn't a circuitous two mile hike with intermittent sidewalks.
 Before they were largely converted to glorified open-air-malls for the cul-de-sac dwellers to drive to on the weekends.
Count up the glorified open air malls in your area and do a little local history.
This can be casually done by noting average home build dates in various neighborhoods. Find a neighborhood with houses from the 40s/50s and note the layout. Then find a neighborhood built in the 70s and note the layout. You can do that with any number of real-estate tools and/or Google Maps.
Just in the county I grew up in there was at least a dozen of those small towns -- each with shops, mainstreets, mixed-use zoning, surrounding homes, often with train stations and/or bus depots (some surviving to this day), all very walkable and livable.
Then the farmland around them was cut up into suburbs in the 40s and 50s.
But even those were notably different from modern suburbs: walkable, smaller lots, still some mixed use with corner stores and restaurants. The cul-de-sac and single-use zoning atrocities didn't start showing up until the 70s, when they basically stitched the old towns and old neighborhoods together into one largely-unbroken stream of Modern Americana.
And again, the thrust isn't that we should all have small towns or that suburbs are wrong for everyone. It's that the old spectrum has been reduced to a binary.
You have suburbs and you have New York City.
The fact that 'US city' discussions always revert to talking about a handful of the oldest cities is itself a tell: if it didn't grow into a city before suburbanization, they didn't really grow into a city. Many look for all the world like a high-rise downtown bolted onto suburban sprawl like an upscale version of a big box strip mall. They don't really "count" as a city like New York or Chicago and we all know it.
Don't know where you live, but we still have boatloads of small towns here in Michigan. Farms and rural townships too. As far as I can tell from my travels, the same goes for the state of New York, too. (And Ohio, Wisconsin, Indiana...)
The small towns exist, some moreso than others, but (at least the ones to which I refer) they don't really exist in the same way. The shops and amenities and zoning that used to make them work aren't there in many places.
In how many could you walk to a local grocery store, rather than having to hop in a car and drive several miles to a 'supermarket' or outright to a big box store? How many have a local hardware store, so you don't have to hop in a car and drive? How many have the apartments and lofts that make the place a viable neighborhood for people who can't/don't have a car? How many have bus routes to get you across town or to the next town? Many still have restaurants and small shops, but even the small town movie theatres are pretty much extinct.
Thank you for conceding they do indeed still exist.
Of course they don't exist in the same way. This is 2013. What you're describing is some sort of semi-mythical pre-WWII world.
Technology changes everything. TV killed small movie theaters. Netflix, etc. killed video rental stores. Amazon killed brick-and-mortar bookstores. Ubiquitous cars killed passenger trains, and meant putting the big shops on the outskirts of town made more sense than putting them in a smaller building right in the middle of downtown.
And guess what? If you live in a small, rural town, pretty much all those changes were huge net positives. (Presuming, of course, that you have a car, a TV, and Internet access.) Sure, you don't have a local theater anymore. But the local theater was the only way you could watch a moving picture of any sort back then. Now you've got 200 channels of TV and cheap streaming movies on a big screen right in your living room. That supermarket you disdain probably has 5x as many products as the old grocery store downtown did, including entire categories of food they probably never dreamed of carrying, and better prices to boot.
All that said, most of the small towns I'm familiar with have a grocery store within city limits -- and they're small enough places that means they are walkable, at least in the summertime. Indeed, every place I've lived since leaving my childhood home has had a grocery store within walking distance, even though I never even vaguely considered that a factor when considering apartments or houses.
Some small towns definitely still exist, but they're the minority exception instead of 'the way'. That's all I was ever saying. You used to have dozens. It used to just be the way that population centers grew. Now you have a handful of conspicuous exceptions.
And technology has little to do with this change. Plenty of places are still trying to run bus services. Because ubiquitous cars still aren't. But all the intervening challenges have made that near impossible to do (cost)effectively.
Similarly the theatre is gone because once zoning presses you into a car to get there, what's the real difference between 5 minutes to the local downtown or 15 minutes to the multiplex at the mall?
The root cause was largely a socio-political failure. Self-segregation, myopic zoning, a belief in perpetual growth all abetted by enough wealth on the part of the builders to not care about long term efficiency.
I think this is the heart of the matter. Why are people disengaging?
I have the same experience as you in central Austin which, while not NYC, is still one the biggest cities in the US. I encounter people I know everywhere I go. The grocery store, theater, bar, etc. But I'm engaged in my community.
I think there's something else at play here. I suspect many people aren't interested in engagement and are seeking out ways to retreat. Clearly in your rural Texas town someone could still buy a house miles away from anyone and not engage at all. I wonder if people aren't doing so, or if they do so in such small numbers that you don't really notice.