To some extent you are correct. But that is not the only component. There are requirements in place, from the Government, that raise their costs. Artificial restriction on the area provider can seek clients, for one. Another issue is the customer not paying anything, that leads to "who cares the cost I don't have to pay". That is a big fault in all pervasive insurance. In the past I at least got a copy of the bill that went to insurance, now I see nothing. Comparison of administrative cost between Medicare programs and commercial programs is also an unfair comparison as that do not follow the same rules. Medicare does not REQUIRE a profit. Medicare gets to decide, on its own, how much it will pay. The medical provider is prohibited from asking the patient for the difference.
"Administrative costs account for 25 percent of health care spending, but little is known about the portion attributable to billing and insurance-related (BIR) functions. We estimated BIR for hospital and physician care in California. Data for physician practices came from a mail survey and interviews; for hospitals, from regulatory reporting; and for private insurers, from a consulting company. Private insurers spend 9.9 percent of revenue on administration and 8 percent on BIR. Physician offices spend 27 percent and 14 percent, and hospitals, 21 percent and 7–11 percent, respectively. Overall, BIR represents 20–22 percent of privately insured spending in California acute care settings. Single-payer analysts Steffie Woolhandler, David Himmelstein, and their colleagues have argued that moving to a Canadian-style system would reduce U.S. administrative costs by 10–15 percent of total health spending." That means instead of 20% it would be 18%, big savings!