[I'm hoping this education example will give some insight into the public health care plan, or at least give you another way to think about it.]
Suppose that education is only available from private sector schools, and that education within this system is very expensive. Because of the expense, millions of people do not have access to education. Further suppose that due to the characteristics of the education market, there is reason to believe that the private institutions are bloated with excess costs (and, in addition to all the other excess costs, 30% of their expenditures came from competition for students rather than delivering education). To make matters worse, the already too high costs are expected to escalate rapidly in the future and further limit access to education. (And there's more. If costs aren't controlled, the government's Educare program for the very young will begin to eat up a huge share of the federal budget.)
Now suppose the government decides to solve both the access and cost problems by setting up a public plan for education. Here's how it works.
The government will build schools, staff them, purchase supplies, and so on, but there's a catch. The schools will have to run without any government subsidies, none at all, not a dime (so this is different than what we actually do since some or all of the education bill is subsidized, some for college, all for lower grades).
If these schools provide exactly the same education as the private sector schools but cost less to attend, then that would either force the private sector schools to find a way to compete by bringing costs down, and they ought to be able to match the government run institution, or they would go out of business. It's true that the public institutions might have an advantage in buying books in bulk, that sort of thing, and they could probably get books and other supplies for less than individual private schools could get them, but what's wrong with scale economies? And to the extent that it is the power that comes from their size as public institutions rather than actual efficiencies, it's important to remember that the publishers aren't without their own countervailing market power, so this makes the playing field more level.
As to access, one option is to do as we do with schools now and implicitly subsidize everyone who attends, rich and poor alike, by giving government subsidies to the schools (tuition falls by the amount of the subsidy, to zero for public elementary and high schools, part way to zero for colleges). But that would violate the no government help rule we imposed above. The other way to do this is to take the money that would have been used to subsidize the schools and instead give it out to individuals who couldn't attend school otherwise (perhaps graduated by income). That avoids giving subsidies to those who don't need them, and the subsidies can then be concentrated on those who do. The additional help available to those who need it would, in turn, allow more people access to education, a key goal of the policy.
So, the idea is to build government schools that must run without any help from taxpayers, and the public schools will compete side by side with the private schools. Rather than limiting choice, this adds one more choice, and it's a choice nobody has to make if the public schools turn out to be more expensive than than the competing private schools. Then, to increase access to education, give individuals the tuition subsidies they need to make it possible for them to attend the public school. Finally, for any conservatives opposed to the public plan, notice that if individuals can use the subsidy on either a public or a private sector school, this is basically a voucher system. However, in this case the goal of the voucher system is to reduce costs in the overly expensive private sector rather than to discipline the public institutions, something the private sector shouldn't fear if, in fact, it is the least cost provider of education.