The insurance is private, the healthcare providers are private. Health insurance is mandatory. That pretty much describes Obamacare, except the Swiss will sue you for not having the mandatory health insurance.
I mean, there's still a difference. The profit motive is now fully normalized in US healthcare. At this moment, just about every single hospital in the US is understaffed to the point of patient endangerment / harm. And nobody can do anything about it, because it is the entire industry, and our society has accepted that maximized profits come before patient care.
“Patient care” is not the ultimate good. It must trade off against other priorities in the society. We cannot dedicate 100% of our resources to patient care.
This means that even public healthcare systems must operate on a budget, the resources are still limited, and in actual practice this means that the public systems are also understaffed, often (if not, in fact, typically) more so than in US.
In my personal experience, wait times and availability of healthcare in public system in Poland are pretty dismal compared to US. I also heard similar opinions from most of my friends and colleagues who immigrated to US from Canada and UK; they universally say that they are actually able to get higher quality service in US with shorter wait times.
You cannot just throw catchphrases about “maximizing profits” and proceed as if it settles the issue, you have to actually argue that some other approach will bring better outcomes, and you need to argue why similar approaches to what you propose applied elsewhere have empirically not brought the outcomes you expect to see in US.