If you manage to find a supporter of nationalized health care to debate, and you manage to move them past "fascist dupe," they often point out that we have rationing, of a sort, now. Health insurance companies won't pay for certain procedures on the basis that they're not cost effective, or are experimental, or some such. "Wouldn't you rather," they say, "have these decisions made by government."
"No, I bloody well wouldn't," you respond. Correctly. But why wouldn't you? In addition to our well-founded suspicions of government, which the left share in most other situations, there are several tangible reasons not to want to fight with the government.
Now, if you and your health insurer disagree about a treatment, you go through a relatively simple internal appeal, they deny you coverage, and then you sue them in federal court. (You get to go to federal court because most people get their health care through their employer and employee benefits are regulated by ERISA, a federal law. If you buy your insurance directly and get to go to state court, that's even better for you and yet another advantage lost to nationalized health care.)
If the government denies you coverage, you get to appeal, in the first instance, to an Administrative Law Judge. Despite his title, an ALJ is not a judge. He is an executive branch bureaucrat, answerable to the same agency hierarchy that made the decision to deny coverage in the first place. Now, so far this isn't too much different than the private sector internal appeal. But there are differences. First, ALJ proceedings, in my experience, take longer than internal appeals. Second, health insurer appeal panels tend to include practicing doctors, so there's some chance of a favorable verdict. ALJs will be lawyers, not doctors, beholden to the agency for reappointment (they serve for a fixed term (6 years, usually) and not for life). So there's somewhat less chance of a favorable result.
But the real difference comes when you go to court. Now, we don't know what the final statute is going to look like, so I'm going to assume that you will be able to appeal the ALJ's decision to the district court for determination de novo, that is, without the court deferring to the decisions of the ALJ. If you have to appeal to a federal appeals court, or if the court has to defer to the findings of fact of the ALJ unless they are clearly erroneous, then you are sunk. If you sue a private insurer, you just have to prove that your version of the facts is more likely than not.
If you sue your private health insurer, you'll go in front of a federal judge who, probably, doesn't particularly like insurers. In other words, a lawyer. Even if he doesn't hate insurers, he is the judge; he won't want to defer to the company. (Does ERISA suggest that the judge should defer to the company's denial? Arguably, but I've done a lot of ERISA work and, believe me, they don't.)
If you sue the federal health provider, on the other hand, the judge -- a federal employee -- is now meant to second-guess the workings of other federal employees. Is he going to treat them the same way he treats insurers? No way. In fact, he shouldn't. Even if the judge doesn't have to defer to the factual findings of the ALJ, he must defer to the expertise of the agency within the scope of its mandate and to its legal interpretation of its own regulations and establishing statute. So if the agency says that, as a matter of law, grandma doesn't get her hip replacement, then the judge must accept that decision unless it is arbitrary. Which it never is.
If you sue a private insurer, you can argue that they are estopped from denying coverage. That is, that they told you there was coverage in this situation, you relied on the statement to your detriment, and now they're not allowed to change their mind even if the first statement was wrong. You can never estop the government. (There is an argument under ERISA that you can't estop insurers, but I don't think that's the prevailing law. In fact, this is a nice example of how judges give private insurers less latitude than they'd be forced to give to a public insurer.)
If you sue a private insurer, then the court will construe any ambiguity in the policy -- and there is always ambiguity in an insurance policy -- against the insurer. If you sue the government, the court will defer to the government on how any ambiguity should be construed.
Judges have no hesitation to order insurers to spend money. Although they will do it, judges are supposed to be reluctant to order the government to do anything.
Basically, this comes down to the question, would you rather sue a private company or would you rather sue the government. You'd much rather sue a private company.