My hope, perhaps naive, is that 95% of what happens in this Rube Goldberg machine basically amounts to the government paying private insurance companies to pay doctors, which is inefficient but ultimately straightforward.
That's an understatement on the order of saying WW2 was a rebalancing of geopolitical power.
Healthcare is what? 20% of GPD. Likely half of that is paper pushing compliance checking rubber stamping that likely ought not to exist.
Compare that to Swisslife, my previous private healthcare insurance: for every 100€ I gave them, 66-72€ were paid to insure in need while I had them. Of course the mandatory dividends were pretty high before COVID, and reinvestment/marketing/sales has to take a bite of that, so perhaps only half of the 30% they take from customers are admin costs, but still, I'd rather give more to my country national healthcare if I could.
Probably the variance you have could be caused by non-admin costs being counted as admin costs.
22 % Social Security
14 % Net Interest
14 % Medicare
13 % Health
13 % National Defense
10 % Income Security
5 % Veterans Benefits and Services
2 % Education, Training, Employment, and Social Services
2 % Transportation
1 % Natural Resources and Environment
3 % Other