Snoopy, I think the difference is that your children are affected and you are driving on their behalf. Indeed when mine were young I took them - the first anyhow - to doctors in NYC when special problems arose. I myself went to NYC or to the centers you mentioned when a) I was younger (and my scoliosis wasn't as bad) or b) someone drove me.
Now, though, I live alone and am unable to drive for more than an hour because of severe back pain. Even being a passenger is hard. And if I fly or take a bus (or train) - it's very difficult (and expensive!) to get around. I use a walker. Even changing trains is impossible without a companion and who can afford that?
Even the nearest train station is almost an hour away - an unmanned station with only one train daily departing and one returning . That means for a single appointment, I have to pay for two nights in a hotel and countless tips to maneuver at all - plus many cabs and, of course, hours of uncomfortable traveling to get there and return (public transportation is much slower than car).
I have a number of friends who perished because they relied on our sub-standard local hospital and our (mostly) sub-standard physicians. That was because they couldn't travel easily to a major medical center - even those with helpful extended families!
You are in a more privileged position and it also sounds like you are a lot further East than me!