Billing and insurance ("BIR") is reported at around 8.5% of revenue --- admin isn't just BIR, as you note. But replacing private insurance with Medicare doesn't drive BIR to zero; Medicare is also a claims-based system. Estimates are that you'd cut BIR by somewhere around 30-40% (of 8.5%).
Most of the savings in these kinds of reports simply comes from paying doctors less (or delivering fewer procedures, which is also a problem we have.)
Physician/nurse salary is not the major driver of the cost of healthcare. It's around 12-15% spending, depending on the methodology. Even if you force doctors to work for free, it won't meaningfully affect the cost.
It's really the overhead costs that are so Byzantine that they can't be quantified properly. Hospitals have teams of coders, insurance companies have teams of counter-coders, physicians have to waste their time on calls with insurance companies, etc.
"Medicare for all" would alleviate a _lot_ of this. It already works for the elder population, and pretty much every senior has health conditions. So extending it for everybody would result in savings. This is a no-brainer from a purely fiscal point of view.
Another way to fix the mess is to lean on the free market side: prohibit employer-sponsored insurance. Completely. All the health insurance plans must be sold on the open market to everyone.