For starters, American insurance has a "maximum out-of-pocket" amount, which means the maximum you can possibly pay for healthcare costs. My plan, from just a regular unknown company doing boring things, has a maximum out of pocket of $5k for an individual. So there's no scenario where I'd ever benefit from spending "a couple tens of thousands of dollars" because even if I spend the whole year in an ICU bed at a cost of millions of dollars to the hospital, I only pay $5,000.
Also, "a lot of people either bankrupt themselves, and end up paying much more than that" doesn't make sense. Declaring bankruptcy means you don't pay the debt, you wouldn't pay a lot AND declare bankruptcy. You'd see the amount was too much to pay, declare bankruptcy, and have the debt wiped out.
Keep in mind that millions of Americans have essentially no assets that aren't protected in a bankruptcy (car, home and retirement accounts are generally safe). It's not like millionaires are going bankrupt from medical costs, it's people who had nothing to begin with declaring bankruptcy when they got hurt while uninsured and going back to zero (instead of negative).
The real problem of the US system isn't the subsidies for the poor, it's the opaque, convoluted billing system between insurance companies, pharmacy benefit managers, and providers. Billions of dollars are siphoned out of the system as profit to insurers and hundreds of millions are wasted on salaries for the bureaucracy of managing the billing system.