That's mostly correct, except that there are three classes of payers: Medicare/Medicaid, cash patients, and insurance. They form three separate supports that must together cover the expense.
In practice, Medicare/Medicaid form a near monopsony -- they're the 800lb gorillas, and can demand capitulation to their reimbursement rates. On the opposite side, cash patients are few, and many of those are expected to be bad debts, and so contribute little. Thus, the pressure release must be through private insurance payments. And that's why your insurance rates are increasing.
This is also why many experts fear that Obamacare's plans will squeeze private insurance out of the business, despite the "public option" being putatively off the table. If regulations hold government reimbursement down to a fraction of real cost, private insurance is forced to take up the slack. That necessitates raising rates, and thus forces customers out of private plans. But they're required to be insured, so they must move into a government-controlled plan.
A proper and accurate accounting is important. We should know how much of the cost of treating patients who don't pay is covered by the hospital, the private insurance companies, the taxpayer, etc.
In terms of the argument about the "public good", though, I think the original point is unaffected. You wrote that private insurance "is forced to" take up the slack. Who is forcing them to treat these patients? Medical ethics does to some extent, there is what is called an "imperative to rescue". But there's also the long arm of the law - an ER that refuses to treat a critically ill patient may face criminal charges and other sanctions. Private hospitals definitely do transfer these patients to county if they can possibly get away with it, but at times they must treat and receive medicare reimbursement that doesn't cover the cost.
So we actually have a public health care plan. Right now, our public health care plan is a patchwork of regulations that encourages people with no money and no insurance to use the ER for primary care, resulting in a burden on taxpayers, private insurance, and private hospitals. A lot of people, myself included, believe that this has resulted in an extremely expensive and ineffective system of quasi-public health care. This is why I would support some form of organized public health insurance, in spite of the fact that I tend to lean toward market solutions and small government as a general principle.
In fact, a recurring discussion between my wife and I includes me asking her these same questions ("Who is forcing them to treat these patients?")
I'm generally unsatisfied with the answers. A legitimate part of it is that it's a Catholic hospital, and so feel a greater responsibility to help.
But there's also an aspect of just needing to play along with the regulatory game. It seems that they've got themselves so dependent (or is that "addicted"?) to governmental aid, that they can't see a way to continue without it -- and that means full participation in all the wasteful, inefficient stuff. I believe (without quantitative proof) that at the bottom line they lose more to the bureaucratic garbage than they gain in subsidy.
In practice, Medicare/Medicaid form a near monopsony -- they're the 800lb gorillas, and can demand capitulation to their reimbursement rates. On the opposite side, cash patients are few, and many of those are expected to be bad debts, and so contribute little. Thus, the pressure release must be through private insurance payments. And that's why your insurance rates are increasing.
This is also why many experts fear that Obamacare's plans will squeeze private insurance out of the business, despite the "public option" being putatively off the table. If regulations hold government reimbursement down to a fraction of real cost, private insurance is forced to take up the slack. That necessitates raising rates, and thus forces customers out of private plans. But they're required to be insured, so they must move into a government-controlled plan.
Edit: awkward phrasing in 1st para