The taxes in France presumably pay for everybody, while the US premiums only pay for similar healthy working people.
Healthcare is expensive. The US government started providing healthcare to old people, rather than letting them die on the street outside the hospital.
But rather than the government paying the full cost of this program (ala food subsidies), the government dictated that providers give them special lower prices. The hospitals want to make up for that money some how, and so attempt to pass on extra costs to others. The next target (private health "insurance") doesn't want to subsidize that cost either, so they negotiate lower rates. The hospitals are still optimistic that they can still fool someone, and so attempt to take it out of helpless patients.
The entire system is essentially musical chairs. This is why after you receive medical services, the hospital will send you a fraudulent bill with a fictitious inflated price list rather than waiting for your "insurance" to cover their part first - they're hoping to luck into a sucker who will mindlessly pay the sticker price rather than the back room reduced rates.
I can foresee a trend of people setting up trust funds for themselves, to reduce the chance of losing their assets to the healthcare reverse lottery. That's really the point we're at - the last attempt at healthcare reform was basically jiujitsued by the health "insurance" industry into making patronizing their cartel mandatory!