Well, I try not to go around annoying people about my country and its social organization, so I just tend to shut up, but I just can't resist it this time.
iamzack, I'm sorry to disappoint you, but it just doesn't works. Argentina has free health care paid by the national, provincial and municipal government, health care and covenants with private institutions provided by the unions, and private health care. The result? Mediocre public health care paid by the middle class through taxes, unions getting incredibly rich and bureaucratic, and private health (generally the only that provides top of the line, state of the art health care) facing an unfair disadvantage in the market and thus becoming more expensive.
Why do we do it this way then? Well, just like with education, it's the only way we can provide those services for the considerable part of the population that simply can't pay for it (when it comes to human rights, you can't simply let them die). That's not your case. You guys earn enough, and you have a reasonable system working right now. If sometimes you have to make some sacrifices to save money for a surgery or something, the best solution is simply doing it (I'm in the middle class, covered by the public system and three unions and I still had to do it anyway for my past three eye surgeries, and for the next three too). Giving more money to the bureaucrats won't do anything to improve the situation, and it has a good chance of worsening it.
You're fine the way you are. Don't fix what's not broken.