It's not just Switzerland and France; the Netherlands also has a private-only health insurance system. It's also very difficult to draw decisive conclusions since, across countries, there are hundreds of confounding variables — it's not just public vs private, but it's also which regulations exist in each country, whether it's employer-sponsored vs individual, general willingness to pay, etc. You're correct that Switzerland has a system like the ACA in the US, but the biggest difference is that it's not common for the Swiss to get their private insurance from their employers; it's all on the individual market. The US is actually unique in that regard, and is probably the most significant difference — the vast majority of working age adults in the US get their insurance from their employers, and as a result the ACA's individual market has been in a dire state since the program's inception.
Also "socialization" is very different from "nationalization". The general trend you're talking about is more to do with the fact that having society subsidize healthcare for the poor can lead to better outcomes. As it relates to who actually does the insuring, underwriting, and payment (public vs private), one isn't necessarily better than the other; each has its trade-offs. It's just that the US (in particular) has chosen the worst of both worlds.
I work in this industry, and from where I sit, the closest thing we have to a clean A/B test that controls for all of those confounding variables is actually being run in the US right now, with Medicare. When you turn 65, you have the option to enroll either in "Original Medicare", which is what we usually think of when we talk about "single payer healthcare in America", or you can enroll in Medicare Advantage (aka Medicare "Part C"), where the premiums that would go to the CMS instead go to private insurers like Humana, United, Oscar Health, Aetna, Clover, etc. These plans replace Original Medicare.
- 48% of Medicare beneficiaries are on private Medicare Advantage plans instead of the public "Original Medicare". Because everyone is entitled to "Original Medicare", this is purely voluntary. This number has been growing so rapidly that the CBO projects that by 2023, the majority of beneficiaries with choose the private over the public option. The CBO further projects this proportion to increase to 61%(!!) by 2032. (https://www.kff.org/medicare/issue-brief/medicare-advantage-...)
- For most beneficiaries, Medicare Advantage costs about 40% less than Original Medicare and are, on average, of higher quality than Original Medicare (https://healthpayerintelligence.com/news/medicare-advantage-...)
- In Urban areas, Medicare Advantage costs less per capita to administer than Medicare — and that's not including the extra Medicare Part D insurance that you would have to buy if you're on the Original Medicare plan (https://www.commonwealthfund.org/publications/issue-briefs/2...)
So no, you cannot look the cost difference between the US and other countries and simply conclude that it's because of private insurance, because the actual data tells a different story. And "universal healthcare" is not the same as "public" healthcare. It might help to think about it this way: universal access to food can be achieved without nationalizing the food industry, or the food payment industry.