Insurance is meant for rare, but disastrous things. Health insurance would only cover surgery, cat scans, things like that. But it would NOT cover ordinary medical bills.
Personally that's what I pay for. It's called major medical, or high deductible insurance. You have it for years and never once use it.
There's probably no chance of the US covering this type of care for everyone. But that's what classical insurance is: pool everyones money to handle a disaster.
Trying to cover people for ordinary care is welfare, not insurance.