This is well-studied and well-known, with one study showing private insurers in Canada having overhead expenses ten times higher than the public system. You can read more evidence yourself, here, here, here, here, here and here. I could go on, but you get the picture.
Interestingly, the very same study also found that overhead among Canada's private insurers was higher than in the US.
The heavy administrative burden of private insurance is undeniable, resulting in a good chunk of the waste in our existing prescription drug system. And it's not just from private plans. Existing government plans also take cues from private insurance when administering public programs. That's why we waste money administering useless policies (user fees or income and condition testing) that are actually designed to deny coverage and raise profit, not run a government programme.
That's a lot of churn, ultimately doing nothing but denying care to people who need it.