As an Australian expat who's lived in all three situations (Australia with single-payer, USA with high level private insurance and Switzerland with government required and regulated), I much prefer single-payer.
Pure private like the US is just insane. It technically works but when you go in you have no idea what you're paying. Random service providers can bill you months later and there's not much you can do to stop it. Plus you have to be sure your provider is in-network and a bunch of other pointless nonsense. Finally, insurers can charge you whatever they want.
Switzerland's eliminates most of these problems. Everyone is required to have a "basic" level of cover, where "basic" is defined by the government to include basically everything medically necessary (in much more precise terms than mine). The maximum premium is fixed to be just a little over what's profitable for a well-run company but customers get to choose their deductible, up to a mandated maximum. All procedures and medications have fixed prices.
Thanks to the regulated premiums and procedure prices, you end up with everyone paying roughly the same amount for the same thing but to different people. Service providers have to send and process bills for 30 or 40 different insurers or hundreds of individual customers. Customers have to pay bills to their insurer and/or each service provider they visit.
This makes no sense to me. When everyone's paying the same amount for the same thing, it doesn't matter that there are different choices. By centralizing everything with one insurer, you can drastically reduce the paperwork for everyone. Providers send bills to one insurer. Customers pay bills to one insurer.