"uninsured people pay way more"
A minor but important correction is they are billed way more, not necessarily pay way more.
For example I dated a girl who just out of high school with no insurance had a really badly broken arm (car accident IIRC) and the hospital "charged" her, in total, $100K+ as an imaginary amount, complained at her for awhile, and then they wrote her off after some token payment (more than they would have gotten in bankruptcy, less than it would have cost her in legal fees to file bankruptcy, but a moderately huge amount of money at part time minimum wage income).
Now whats important is the difference between the imaginary retail price of $100K and what it actually cost to provide services is incredibly valuable from an accounting standpoint. Also from a marketing standpoint her "$100K" of services was donated by a gracious company to the poor, they sure as heck don't publicize the actual cost of providing that service was only $10K even including minor surgery and recuperation. They did a good job, by the time I met her, I had no idea her arm had been so ruined just a year or two ago until she told me and showed me her arm, by no means did they cheap out on her care or recovery just because she was uninsured.
Finally the hope is that if they charged that poor girl a fair $10K then it would be hanging over her head for years until she could pay it off or more likely bankruptcy filing fees costing less than $10K they'd get nothing at all, but if they "charge" $100K then they might jackpot and she could qualify for some govt grant or fundraiser or whatever. If they know she's going to declare bankruptcy no matter what they charge and they're going to get $0 revenue, then game theory shows they should ask for $100M on the off chance by a miracle she wins the lottery, or qualifies from some weird govt grant or school support fund or something.
I think something foreigners often don't understand is we charge poor people $100K for a broken arm but we only charge them like $2K to file bankruptcy, so there are no acute medical problems that cost more than $2K, and for chronic problems they make life hell until either the chronic problem goes away and so do the bankruptcy bills, or the victim dies, but its not really expensive. The business model ruins middle class aspirations (how will I have the vacation home and RV if I file bankruptcy?) but we solved that problem by shipping off all the middle clss jobs to China, not having a middle class means not having to worry about a middle class anymore, there's just really rich people and peasants who own nothing.