You missed the biggest and most expensive reason.
We have socialized provider system, but a private payer system.
Anyone can get care in the US, no matter their ability to pay, especially with emergency care. The cost of providing this care is passed on to those who can still afford to pay for medical costs or insurance.
There are a huge number of people who make to much to qualify for government programs, but not enough to afford private insurance and are not provided with insurance through an employer or do not qualify due to pre-existing conditions. The number of people in this category at last estimate is around 45 million. These people still use healthcare services, a few can pay for it out of pocket, but many end up getting the bill forgiven through bankruptcy or other means.
If all these people became payers into the system, at any level, it would bring down costs for those who can still afford to pay for insurance. It would not bring down the overall costs as much as other measures, but it would spread the cost for a health care system much more fairly. It can bring down costs in that people will be more likely to seek cheaper preventive care instead of waiting until a problem requires much more expensive emergency care.