Okay, you've got a couple of problems here. First:
Healthcare is full of fake non-profits. They're NP because they "donate" care to the needy - though in reality they are required to treat everybody by law,
What you're thinking of is EMTALA (Emergency Medical Treatment and Labor Act). It's the big dog in the "you have to provide medical care" laws (in the US). It basically boils down to "If you want to be eligible to be paid by Medicare, and you have an emergency room, you have to evaluate anyone who shows up at your emergency room to determine if they have a life-threatening condition, and stabilize them if they do, as well as providing obstetrics services to anyone in imminent labor." If you show up to the ER with a cold, they don't have to treat you, they just have to evaluate you to be sure you're not gonna croak, then they can kick you loose. Now, the reality is, the evaluation is basically the hardest, and most expensive, part of an ER visit, so they tend to tack on treatment, but they're not required to. (Also, the prevailing attitude in emergency medical workers is that we SHOULD treat everyone, billing can worry about whether or not we get paid for it.)
Nearly every NP hospital has a team of lawyers which clog the judicial system chasing payments from non-paying patients.
Here, I can only speak anecdotally, First, my experience as a patient: I recently broke my leg badly enough that I required surgery, and I have no insurance. I could have gone to the university hospital, which receives funding from the state and the county to take care of people with no insurance, but for various reasons, I preferred to go to the local non-profit hospital. I went to one of their financial counselors, explained my financial situation, provided them with a copy of my most recent w2 and my last two paystubs, and they qualified me for 100% charity care, which means they're not going to charge me for anything, which by the time I'm done with everything is going to be around 35k USD that they'll be picking up. (As an aside, I have a job I had to go to college for, work 48 hours a week at, and am open to massive liability every day I work, and I make little enough money that I qualify for charity care, thus the pay scales in EMS.)
Next from a professional perspective: I work for a company that does a lot of inter-facility transfers, both taking people back to nursing homes, and what I personally tend to do the most of, since I'm a paramedic, is taking people from the local small community hospital (which is a non-profit), down to the university hospital, the big non-profit I went to, or a big for-profit, basically because the patient needs better medical care than the community hospital can provide. It's all done on a case by case basis, but if for whatever reason it's determined that the pt's insurance won't pay for our services, because they don't have any, because the pt has reached their cap, whatever, the small non-profit hospital will pay us. In fact, to take it a step further, I know of at least one patient recently who needed to go into a nursing home, but he is 42, so he doesn't qualify for medicare, and he didn't have any insurance, so in the interim time while they were working on getting him qualified for medicaid (about 3 months that I know of), the hospital was paying for the nursing home, as well.
As a final anecdote, and somewhat off the topic, but I've always worked for for-profit private companies, and every last one of them writes off bills all the time. In fact, the general rule of thumb is that for 911 services, you're going to write off 80% of your bills.