As far as whether healthcare is a right or a privilege goes, I see it as a right.
Wolves and Cape Hunting Dogs care for their sick. Go figure. In additon, any country that can afford a substansial standing army, navy and airforce to care for its citizens can easily afford to provide healthcare to care for its citizens. No matter what political principles you might hold, Homo sapiens is a social creature, always has been, always will be.
The above is opinion; you don't have to agree with me.
Basically, a free-at-point-of-access system,like the NHS, is cheaper than a private system. There is no way in the world that a private system can charge individuals less than they would pay in tax, as they need to make a profit. The lack of a profit motive can and has caused problems in public health systems, as it can lead to inefficiency. But there's no owner or shareholders looking to walk away with a profit. Thus, logically, a public health system can be more efficient and cost-effective than a private one, but has less pressure to make it so.
However, the disengagement between the individual and their own healthcare funding (i.e. you pay a tax everyone does and get whatever you need in healthcare, rather than having the individual choices you make about healthcare influencing what healthcare you can get) leads to problems. People are isolated from bad choices, for years and years and years.
Thus, you can take a pretty decent Heath System, put a government in power that is opposed to public spending for 17 years, have them cut taxes and cut spending, and 17 years later have a Health System with problems. "Everyone" was all for the tax cuts, but didn't (because of the disengagement) foresee that the tax cuts meant that the Health Service (and Education and the Public Transport) would wither over the years.
It has taken since 1997 when Labour came to power to basically erase the damage 17 years of Conservative government did.
But if it is all good (apart from needing to especialy ensure there is efficiency and having to educate voters that if they vote to cut taxes, services will suffer), why does anyone WANT private heathcare?
Well, there's two reasons. First of all, private heathcare allows for massive profiteering, benefitting a small minority at the expense of everyone. This small minority will lobby like hell to be allowed to have their license to print money, and pay for political support. Second, if you are very rich, you can obviously afford to have better heathcare than average (even if average is world-class, it can stil be better).
So, if you're interested in yourself and your neighbour, and you are not wealthy, a state-organsied free-at-point-of-access health system funded by taxation is the best solution.
If you're in the medical business, or are wealthy, a private heath system funded by individual contributions (or those of employers) is the best solution.