I never knew someone in the US that had to get cash to the doctor on the side or bring their own toilet paper and food to the hospital.
I had a friend who broke his arm. The emergency room doctor wouldn't touch him until the family provided some cash.
If you have to provide extra bedding, food and toiletries for a hospital stay on your own - those costs aren't reflected in the taxes that one pays for health care.
This has nothing to do with debating the merits about either system, I'm purely talking about trying to accurately compare the costs.
The system you describe is obviously corrupt, and I sympathize with the frustration, but I might prefer a system where I had to spiff doctors a couple bills to get an arm set over a system where I could be both insured and bankrupted by minor surgery that happened to snag a loophole in my policy. That's the private market insurance system we have in the US.
I lived in the US most of my life. The health care system there is pretty messed up - in my opinion. But when I moved to Hungary I realized I'd been quite naive about how things are here.
I assumed it was a rather cut and dry fact that health care would be better here - and it isn't as simple as I had assumed.