The Netherlands and Switzerland both have functioning private insurance systems. There are also several functioning hybrid systems, like Germany and Israel.

But the primary reason you don't see more successes is the lack of attempts. It's not as if there is a long list of countries with failed free market healthcare systems, either.

The Netherlands does not have a private health insurance scheme. Or rather, it does, but all the prices are basically fixed, and the biggest choice you have is whom you are paying money to. The services are ultimately much the same.

Swiss here: Our health insurance providers might be privatized, but they're strongly regulated and every Swiss citizen must have healthcare. They do function, but not because of their free market incentives (my opinion), but because of strong regulations. Also, if there was a vote to go from public -> private insurers again (historically, they were public once), I think they'd stay public, because we're not really seeing the benefits of their privatization, only the downsides (tons of ads, calls, arcane coverage rules etc.)

Here in Israel the majority of large health providers are essentially contractors, are largely indistinguishable and don't compete on price at all. Everybody pays a health tax as a percentage of their income. You don't have to pay anything aside from it, but if you want additional coverage like improved dental and discounts for non-essential medication, the best available coverage through the largest providers would cost you about $25 a month on top of the health tax.

I don't know in what meaningful sense Israel is a "hybrid" system, it functions almost exactly like a 'Medicare for all' type proposal would.