Every advanced nation on earth is a "private option" for every other nation. For example, consider this story:
https://www.fifa.com/worldcup/news/heartache-for-injured-striker-lee-15875
That's about a striker from South Korea's national football team flying to Germany for surgery. He had the money to fly anywhere and get treatment, so he could have picked the US, but picked Germany because that's where the particular specialist considered the best for his condition works.
Wealthy Americans will also, sometimes, travel abroad for healthcare, when the specialists they want are elsewhere. For example, Kobe Bryant famously had his knee treated in Germany. Peyton Manning, A-Rod, and Andrew Luck also all have traveled to Europe for medical treatment. For those with money, the planet effectively functions as a single common market for healthcare, with people going wherever the best specialists, clinics, or experiments for their particular problem are located.
However, what sets the US apart is that the "private option" is the only "option," at least for those who are under 65 and haven't earned government care by way of the military, etc. By comparison, in other wealthy nations, you have a choice of paying out of pocket or getting high-quality care at low- or no-cost, within the government run, or heavily-government-controlled public system.