I tried to explain to people in The Netherlands this exact problem when they were thinking about switching from single-player to a commercial insurance company model.

Insurance companies always have an incentive to make healthcare services expensive. They have even more incentive to make healthcare expensive and do deals on the backside where their suppliers give them a kickback. Even better if said kickback comes via a side door.

So we went for the commercial insurance companies. It took them 4 years before they change the drug choice from decentralised (Pharmacies deal with drugs companies, received small discounts which funded a good quality of care) to centralised (Insurance deal with drug companies, receives kickbacks) and the Pharmacies funding was drastically cut leading to worse quality of care and more drugs being used. Double-whammy because in the previous system the Pharmacy did medication reviews which almost always result in a reduction in drugs (the thing with drugs is that quite a lot of them are given to reduce side effects of other drugs, the original drug gets cancelled but the side effect reducer gets forgotten and just continues ad infinitum).

Health insurance has been super inflationary since then when controlled for quality of care.