Germany doesn't have single-payer, but it does have a highly-regulated, mostly public health insurance industry. Most people are covered by the public system, which, while multi-payer, has premiums set by the government and based on income, as well as a fixed set of services determined by the government.
In effect, it's not that different from a single-payer system funded by a progressive income tax. I have thought the German system might be a better fit for the US, since sickness funds can still compete with each other, mostly on customer service and the like.
All the best systems are single payer. Italy, France, Spain, the UK. The countries with private insurance like Germany are middle of the pack. Clearly single payer is the superior option.
Having lived in both countries, I wouldn't say that the UK system is in any way better than the German system. Rather the opposite.
In addition to that, the German system isn't really a private system. The premium and health coverage is set by law and the insurances are not-for-profit entities. There's the possibility to get private insurance for parts of the population, but you're not dependent on private companies.
Back that up with numbers please. At best, the former four are better in terms of value-for-money, but with experience living in both the former and the latter, the German system is superior in terms of outcomes, waiting times, choice.
(Yes, mediterranean countries have a higher life expectancy due to diet and physical activity, but that wouldn't change much with a different health care model.)
My primary contention is that Every other first world country? is a little too lacking in nuance, I wasn't trying to express an opinion about what works best.