I think the biggest problem which I encountered in bigger cities is the possibility of getting sued. I mean on rural road in Jersey you can pull out a farmer from under a cow that broke his pelvis and femur, stabilize him, drive him to the hospital, save his life and he will be thankful for life (plus you get to marry his daughter and have dairy supplies for life). You do the same in NYC and farmer would sue you for leaving the cow unattended and causing her psychological damage which produced half a pint less milk per year and costed him one billion dollars in income. But he will settle for everything you have.
When I get to the accident scene and I'm off duty I will secure the scene check for life thratening injuries (which I will treat) but rest is up to the arriving police and EMS crew. No point in risking everything you worked hard for (maybe that's the reason why most EMS members keep their assets under their parents' or spouse's name).
Matt