So, some paperwork is mandatory, no actual health care.
The reason many employers voluntarily provide an actual health insurance (or simply reimburse private-sector medical costs directly) is that it is one of the few tax-free benefits that employers can give to people, so it works efficiently as an additional compensation. And of course, because employee absences are costly, it makes good sense to try to keep them in good health.
The public sector health care does practically nothing in terms of preventive health care for people between 25-65 and in employment (except for maternity-related things). This is left to employers.
The way the law is brought to action is that the private clinics have specific service tiers which they offer to employers, of which even the basic one covers consultation by a general practitioner. The easiest way for employers to fulfill the letter of the law is to buy the service.
Which means that working adults seldom tend to the public services unless they are really ill, which means more business for the private clinics and increases the appeal of a health insurance (it's easier).
This is getting really obscure for this forum, but as a reference: The specific law is
https://www.finlex.fi/fi/laki/ajantasa/2001/20011383
Which in google translate is
https://translate.google.com/translate?sl=fi&tl=en&js=y&prev...
Yes, this is obscure for this forum (perhaps that's why some people downvoted the previous comment) but just for the note, in Finland the employer is not really required to buy any services from private clinics. The legal requirement is for existence of a preventive plan, and a specification for what other services the plan covers. It's okay by law if the coverage effectively evaluates to nil, and employees just are off to the public system's queue.
I know in some other countries, like Germany, it is set up much more closely to insurance, in the sense that procedures have prices, are billed out to patients, who are covered by insurance plans that reimburse patients for costs, etc., in which case I imagine the bureaucracy is much more insurance-like.
Private clinics which are the main receivers of insurance money are often owned by the doctors. Thus, they have an incentive to over-charge, over diagnose and run expensive labs whenever they can. I'm not saying it happens, just that the incentive is there. Government provided healthcare offers no financial incentives for practicioners to over-diagnose - on the contrary this system then incentivizes to be as prudish as possible with the public resources.
I'm not saying one or the other is better or worse, just that they are not equal in every way.
It also has voluntary private health insurance.
What you probably are referring to is that USA has not private health insurance at all.
In the US only your employer could give you insurance, so it is not personal and private, but collective, if you try to do on your own it is prohibitively expensive.