The big-medicine industrial complex in this context is not the insurance firm, whether public or private. It is the biomedical firms that are developing expensive new treatments costing five, six, or seven digit dollars per patient per year, and these firms very much would prefer if you didn't take steps to catch conditions early or prevent them cheaply.

Those firms have no say over which treatments are preferred on which patients in single-payer health systems. Actually, it is the exact opposite. When there is a single payer, they can haggle on behalf of the entire population so they have huge leverage over the treatment costs. The government can simply say "You either sell at the price that I want, or lose access to the entire market in this country.". That's why American drugs can be 10-100x more expensive than the rest of the world.