Call it Universal Health Insurance
(Note: This is all opinion and personal knowledge, I’m not an expert and may have some things wrong)
Insurance companies work by having many people who aren’t making a claim pay for the few people who are. Spreading the load. That sounds inherently socialist. That’s a dirty word to some, but in many ways it’s true. Insurance is many people paying into something to help a few. The ownership of the problem, people needing to make claims (car accidents, fire, health, mortgages, whatever), is shared by everyone (the claimant as well as everyone else paying into premiums).
Insurance companies are more profitable when more people pay premiums but don’t file claims. Put another way, premiums could be made cheaper when more people are paying into the pool (assuming profit isn’t a goal, which it is for all private insurance companies). Economies of scale and all that.
The most efficient insurance company is therefore a monopoly. If every single person in a country is paying into one insurance provider, that insurance provider could stand to make the most theoretical profit (or, the cheapest possible premiums).
Now, we don’t like it when companies are monopolies. It’s too risky, they have too much control. So when there’s such a strong need for a service that it necessitates a monopoly, we sometimes move the burden of owning that service into a government agency. Roads, electrical grids, military, and so on. As a society we’ve agreed those problems are universal concerns, and so our taxes pay for all those shared needs.
In the US there’s an ongoing debate about how to pay for health care. Private health insurance, when it’s splintered as much as the US system is, is inefficient. We can see this by looking at the cost of health care in the US relative to other countries, it’s staggeringly high. The US is paying nearly double versus other countries like Germany, France, or Canada — roughly $9500 per person per year vs $4000–5500 in many other countries.

(graph is slightly old, from 2012, link above more recent)
Given the high costs, you’d think everyone would be in agreement that something needs to be done to offer cheaper services, even just as a cost cutting measure. The two most cost effective solutions are either private national health insurance, or government run health insurance. One is a monopoly, the other is a taxed service like any other.
This also begs the question of whether insurance is the right answer at all. We pay into fire and police services through our municipal taxes. You don’t get a bill after a cop shows up to help you after your car was stolen requiring you to file a claim with your crime insurance provider. You don’t pay a deductible on the first $500 of the crime mitigation fees. You don’t get turned away for a pre-existing crime. The point behind police services is when you need them, they are there, you don’t have to think about how you’re going to pay for them. Why isn’t healthcare the same? Insurance and health care may be fundamentally opposed.
While that may or may not be true, UHC does have a branding problem in the US. It carries baggage and causes some to have vehemently negative reactions. Instead, maybe try calling it Universal Health Insurance.
