This assumes a zero-sum economy, which is not how the healthcare economy works at a national scale.
Hypothetically, competing insurance companies would push prices down. In practice, the market isn't a fair market (even before you get into issues like "People need medicine to live, so a market is a bad tool to parcel out medicine because price the patient is willing to pay is unbounded"), so insurance companies tend to add cost through paperwork, managerial overhead, accounting overhead, and cases becoming more expensive to treat as delays in preventative and early-intervention care cause issues to become chronic and urgent-care issues.
When the payer is the government, the government can say "The price of insulin is 2% above cost of materials" ... and it is. And when the money is flowing from taxes to government to healthcare, instead of employees to employers to insurance to healthcare, a company isn't asking itself "Can we afford to take on more employees right now if it's going to cause our healthcare insurance costs and employment taxes to go up," because the decision of how many people to employ is decoupled from the question of how much in taxes the company will pay.