Insurance companies do not want single payer insurance, as is the case in the UK, among other countries. However, they favor the government mandated oligopoly that exists currently, where the Federal and state governments, through Medicare and Medicaid, set prices for medical services. For the most part, medical providers will set their prices on the amount the Federal and state agencies will pay. Insurance companies prefer the stability of fixed prices so that profit margins can be protected. Many businessmen are more like the Jim Taggart character in Atlas Shrugged, willing to cooperate with and bribe officials, and not like Hank Reardon or Dagny Taggart, who want no interference by corrupt and socialist minded bureaucrats.
Except for those medical providers who will not accept medical insurance, our medical system is corporatist, similar to what existed in Fascist Italy and Nazi Germany. The Soviet Union's New Economic Policy in the 1920s, and its much larger and longer imitation in Red China, are also corporatist. Merriam-Webster defines corporatism as "the organization of a society into industrial and professional corporations serving as organs of political representation and exercising control over persons and activities within their jurisdiction".
What we have is neither free market capitalism nor socialism, but corporatism.