As an American living with and around other Americans, none of whom make more than 40K a year (myself included) but also make too much to qualify for low income coverage through combinations of community, city, county, state, and federal coverage (all of which exist and can overlap or stack depending on the specifics)... none of us have paid out of pocket for healthcare except when services are performed by explicitly private entities for services not considered necessary to preserve life, limb, eyesight, mental health, or communicable disease. Dental health appears to be the only exception.
Every other service is "nice to have" rather than "need to have" and I would certainly argue that many of those "nice to have" services ultimately will prevent the need for "need to have" services, so there is definitely room for improvement in my opinion. However, this trope of "Americans have to sell their home to afford an ambulance rode" and other snark needs to be tempered with the reality that those situations are almost entirely because of people intentionally opting out of (or being ignorant of) the existing systems to prevent such financial burden.
I'm sure many will have anecdotes to the contrary. I'm also sure many of those anecdotes will stem from someone either being unaware of available resources or choosing not to use them for one reason or another.
I also know plenty people who believe they will have to pay for things and therefore they don't get help. They don't pay for services bevause they don't get them, and they don't get services because they think they will have to pay for them out of pocket. That mentality comes (in part) from people who keep spouting the same myth that you posted. Please don't spread that rumor, it kills people.