> The US is actually the only developed nation without a public healthcare system,

That's false on multiple counts. First of all, the US does have a public healthcare system, Medicare and Medicaid. Second of all, there exists other developed nations with fantastic healthcare systems that are driven by purely private systems: namely Switzerland (widely regarded to be one of the best healthcare systems on the planet) and the Netherlands. Even more perfectly fine developed nations operate on public/private mixes, including Germany and Belgium.

> it also has the highest healthcare costs of any nation on earth. Coincidence?

In this case, it is indeed a coincidence. In order to attribute "privateness" to the high cost, you have to show a causal relationship. Unfortunately, there's a lot of evidence that makes it very difficult to draw that causal line: private Medicare Advantage plans are about 39% cheaper than the public "Original Medicare" (https://healthpayerintelligence.com/news/medicare-advantage-...), while also being of higher quality (https://healthpayerintelligence.com/news/medicare-advantage-...). In urban areas, the private Medicare Advantage plans cost less to administer per capita than the public "Original Medicare" (https://www.commonwealthfund.org/publications/issue-briefs/2...).