If you're talking in general terms, assume most of the people in the pileup are SINners. Since they can afford vehicles, middle-class or higher, which means wage-slaves. Which means they either have corporate sponsored health insurance or personal health insurance. Lots of Doc Wagons showing up, maybe a few smaller & unknown private firms.
The biggest issue, though, is jurisdictional dispute. What security corp is responsible for the street on which the pileup occurred? If there is suspicion of foul play, would they be fighting it out with Doc Wagon to retain the "patients" in legal custody until things get sorted out? Would they be assigning officers to go with Doc Wagon employees, riding along to the hospital / clinic to make sure the patient didn't escape post custody?
Assume, in 2072, that there is some contract agreement between law-enforcement and medical contractors, but it's not nearly as friendly and "we're all in this together" as it is in this day and age. After all, unlike modern medicine, Doc Wagon has a direct contract with the individual patient, so has a vested financial interest in doing what they promised to do in the contract. And if law enforcement / security gets in the way of that...
Lots of possibilities here. Just saying.