Why isn't it the privatization (which causes those obscene prices) that is the central issue that people focus on? Wouldn't that target the actual issue more directly?
No, because the mere fact that some aspects of the US healthcare system are private is not the reason why the US spends a disproportionate amount of money on healthcare. As you yourself note in your question, there is private healthcare in other parts of the world that does not have this uniquely American problem of high prices. What is unique to the United States is its conception of health insurance, which despite its name does not actually function the way that other forms of insurance are expected to.
How is insurance supposed to work?
Insurance is supposed to be a financial product that will provide you with money in the event an unlikely but costly future event takes place. It does this by charging you and many other people like you a small premium on a regular basis, and if the bad thing happens to you, you file a claim and the insurance company pays you a lot of money from the premiums other people are paying. The main operating assumptions behind this kind of insurance are:
- The event you are insuring against is unlikely to happen
- The event hasn't already happened
- There are lots of people like you facing similar risks
If you have an event that is likely to happen, then the natural thing to do is charge a very large insurance premium that approaches the size of your loss in advance, because otherwise the insurance company (and by extension, the other people who file claims and need money) will lose money. If the event already happened, then with certainty the insurance plan would need to pay the full amount, so the natural thing to do is to make your premium 100% of whatever that amount is. If there are not a lot of other people like you facing similar levels of risk, then the key thing that allows the insurance company to charge a small premium to pay big claims, that all of you will not need all of the money, is no longer operative and this entire system will fall apart.
None of these assumptions are actually operative in the case of what is called health insurance in the United States.
Political intervention in what is health insurance ought to do has made the first and second assumptions in the US health insurance system non operative. People are and have been encouraged for decades to use health insurance for every health related cost they may face under the theory that certain insurance arrangements in the United States make this beneficial (e.g. the HMO, which I would argue hasn't actually worked out that way). Another term for "an event that has already happened" is "pre-existing condition", something that is an un-popular idea that was recently banned as part of the Affordable Care Act.
The result is a system which, despite being called insurance, doesn't actually function like insurance, but instead acts as a form of pre-payment for possible healthcare services, some of which you and everyone else in the plan certainly will use (you go to the doctor every year, right?). This causes premiums to get really high and everything to become expensive. The natural reaction to having a really high insurance premium (where "really high" could be equal to or greater than rent you pay) is to consider whether or not you really need health insurance, and maybe do without it if you're young and healthy. Which, makes the insurance even more expensive, because old and sick people are definitely more likely to have medical bills than young and healthy people. This is a problem called "adverse selection" and it's why the Affordable Care Act and the Swiss healthcare system have mandates to ensure everybody buys health insurance (though in the US case, the mandate was too weak and has recently been repealed).
I know what you're thinking: this sounds like a giant Ponzi scheme that is in the process of collapsing, why would anyone ever get involved in any of this?
And the short answer is, because many decades ago, someone decided that health insurance benefits as employee compensation should have special tax exempt status, so it became popular to offer it as a form of compensation. When that started happening, people who weren't employed by big corporations wanted nice health plans too. Naturally, they asked their Congressman to make that happen. Then Congress kept fiddling around with the system, with the goal of trying to cover 100% of everyone in the country with health insurance, because that's what was politically popular. Now that all of that is failing because the Affordable Care Act has made the fact that this system is defective obvious (but not the underlying defects of the system, which most people don't actually understand) the new politically popular thing is to blow all of this up and replace it with something else.