I wonder if it's like the difference between taking coach vs. a private jet. The private jet is more convenient and comfortable, but safety ratings are considerably better on major airlines.
In this case I probably would not want a head of orthopaedics to treat a relatively straightforward leg break- I would want a practitioner who has a significant amount of recent experience in this area. The ER doctor likely sees hundreds of these a year, so may actually be the best person. Of course if my case is particularly challenging, I want a consultant with a significant amount of experience- but it doesn't seem right for that person to pushing their skills to the highest bidder, regardless of patient need.
I also don't want a steak with my meal, nor do I care for a fancy waiting room, as they aren't going to fix my leg.
In other words, the (I'm guessing) relatively slim chance there is a complication with the broken leg is not worth it to the wealthy father to take.
In medicine, though, there have been a lot of recent studies recently that have shown that too much care can be harmful (especially in the orthopedic realm). Sometimes the best thing to do really is nothing, but when you're paying 40k a year, my guess is you, and more importantly your doctors, would rarely take that point of view.
That's probably what it is, accidents are more likely in general aviation but still rare, however if you must fly commercial you get to face the TSA perverts every single time you go through security.
The same goes for private jets (which I can't yet afford), or private medicine.