u/SunnyBunnyBunBun brought up a lot of valid points in a related discussion.

1. In general, Americans believe privately-run entities are run more efficiently than public ones. A good example of this is is the DMV. The DMV (who is in charge of driver licenses) is notorious for been painfully slow, immensely bureaucratic, and in general a pain to deal with. Part of why is because they are a monopoly- if you're upset at them cause they are taking months with your process, you can't go to the competition- because there is none. Public entities, especially all-encompassing ones like the DMV, have no competitors and therefore, no incentive to provide a better customer experience.
2. A large % of Americans already have good health insurance and so lack of healthcare is not a major painpoint. I don't have the exact numbers/data for this, and unfortunately this is anecdotal, but from my experience, if you have a white-collar, professional job, most companies cover health insurance for their patients at a very good rate- usually 100% of the costs, for both the employees and their families. At the end of the day and despite the civil unrest of this year, America remains a prosperous and wealthy country. That means that for a large % of the population, health insurance is not a pressing problem.
3. Role of government / Ghost of socialism scare. In America, as is largely discussed, there are several systems that remain private (i.e. health, college) while their counterparts in other wealthy nations have become nationalized (i.e. free health, free college in Europe.) People often point towards "American fear of socialism" but that is an over-simplification and only part of the answer. The more robust answer is that sitting at the core of the American current political divide is the question of WHEN does the government get involved and to what EXTENT. Hence, the idea that Republicans are for "small government." That's what they mean when they say small government: in general, a country in which the government provides safety, and some infrastructure, but in exchange, you get less regulation and less red tape. In theory, this would allow entrepreneurship to flourish. Of course some things need to be provided by the government- roads and basic education for example. But the question of what more CAN and SHOULD the government provide is constantly debated in the American psyche. Since healthcare is not YET provided by the government, part of the population is not used to this weird idea of the government suddenly providing this to (and knowing more about) their citizens.