Requiring people to print out a paper and fold it as if it had been in an envelope doesn't authenticate anything but access to a printer and some imagination. Just removing the requirement would be a better alternative.
I live in a country that Americans would recognize as having 'universal health care', and while the US system is also bad, I would prefer it to the fragile and dysfunctional government monopoly that I'm used to. I would prefer a free market with mutual aid societies, but the corporatist system in the US at least lacks the single point of failure and complete unaccountability of a government monopoly.
The US system doesn't typically have choice either except to get another employer... who probably has the same insurance company anyway.
In addition to not being able to choose insurance company, we also have a limited selection of doctors who we can see, assuming you want your insurance company to cover it... and by "cover it" I don't mean the insurance company pays for the whole thing either, its still just a portion... and usually an inflated price because doctors tend to have 2 rates, one for people with insurance and one for individuals paying totally out of pocket without insurance where the out of pocket rate is typically slightly more than what you'd pay out of pocket with insurance but insurance gets a huge bill (which they are all too happy to send you a mailer on how much money they "saved" you).
> complete unaccountability of a government monopoly.
That is a problem with the quality of your democracy, not with government monopolies. A corporate oligopoly is virtually identical to a nondemocratic governmental monopoly.
Democracy is not a mechanism for accountability. No democracy can produce accountability, that's not what it's for. A company with competition has a degree of accountability because their customers can go elsewhere if they are mistreated. A citizen without a lot of money to spend on political campaigns have very little recourse when they are mistreated by the government. Even when the government breaks its own laws, there is almost nothing you can do between qualified, absolute, and sovereign immunity.
The US and other governments have long histories of medical abuses. I would not want an organization that preforms unnecessary and intrusive surgical procedures without consent[0], falsely told people they have cancer in order to remove their reproductive organs[1], intentionally gave people syphilis, hepatitis, and other diseases[2], pretended to treat syphilis to see what happens when it's left untreated[3], etc. to be my only provider of health care. It frankly seems irrational to want an organization that can act with impunity and seem to abuse every function and responsibility it is given to be the sole provider of anything.
I'm sorry, but I don't believe that any, much less the vast majority of democratic countries are run by people who have a decent appreciation of the rule of law.
There are certainly plenty of states that cannot afford these programs, or that may choose to spend their resources in other ways, but the big powers are more than willing to assist when their interests align. I think the case of Denmark shows that it's very difficult to anticipate when interests will align, because we sometimes don't even know the identities of the people whose interests matter.
