It almost seems like obscurity is the goal with the current system. Basically, job security (for admin, insurance) via obfuscation.
It’s more like a politically palatable Rube Goldberg machine for transferring costs to those (well-insured through their jobs, the Federal government, rich supporting institutions and donors) able to pay.
All healthcare systems have administrative overhead, including completely nationalized systems. The US discourse on health care overestimates the administrative overhead by a large amount. We could remove all insurance company profits and adjust our administrative overhead to be in line with countries with nationalized systems and it wouldn't change health care costs by an appreciable amount.
The discourse also underestimates how much of our health care costs go to our significantly higher salaries throughout the medical system. No politician wants to propose reducing the salaries of doctors, surgeons, or even researchers making new medications. The only acceptable villains are the administrators and insurance companies, but even in this inefficient system that's a much smaller slice of the pie chart than most people imagine.
One of the reasons hospitals operate in this convoluted way and still barely clear a profit is that we require them to treat everyone and attempt to collect payment later. A lot of what you and I pay for health care goes to cover people who can't or won't pay.