I'm pretty sure it isn't assumed. My main example has been the same since COVID, as insurance is probably the business you can compare 1-1 the most.
Public health insurance in my country, in the last 20 years used 6-9% (depending on the year) of the taxes send to them as administrative overhead, meaning that for each 100 euro that you paid for health insurance, 91-95 are used to pay doctors, hospitals and medication. The average administrative overhead for private insurance is around 14%, which makes private health insurance 50 to 100% less efficient, and means that for each dollar you pay them, only 86 are used to pay health services.
I have other examples, but it isn't fair: municipal water VS private water service are almost always less expensive and better tested in my country. Municipality trash collection Vs private trash collection, same. Public junkyard Vs private junkyard, same. But in my area, when privatised those services tends to be ran by the local mafia (Marseille, Nice), which add a lot of overhead, and they were privatised because the local government was corrupt in the first place, which means they were probably inefficient (compared to the services still publicly owned) first, then sold.