Sorry no, you've entirely neglected the fact that private healthcare leads to higher per-capita costs as insurers fleece their clients.Insurance companies aren't taking a little profit off the top, they're taking massive profits compared to their costs.
The idea behind a universal health care system is that in such a system, you get the same or better health care for a lower price. It's not a matter of the government subsidizing your health care, it's a matter of giving increased power to the consumer, to prevent them from being fleeced in a transaction where otherwise they would have no bargaining power. And it works. In 2008 (the last figures I've seen, care of the WHO I believe), US health care cost per person was I think $6000/person, while the next-closest was Switzerland (which has a semi private system somewhat like the ACA) where health care costs were half of what they are in the US (I may have the figures wrong, but the proportion is correct, the Swiss pay half per person what Americans pay).
The long and the short of it is that the free market doesn't work as intended where one party has the choice of paying or dying. Government regulation is supposed to (and in most of the world, does) equalize the playing field so that the consumer doesn't get hosed by the insurance company or the service providers.