There's a huge difference between shifting the payment risk from private insurers to the government, and real socialized medicine. Under most socialized medicine schemes the majority of healthcare providers become government employees.
Many, but certainly not all. I imagine most people don't really care who the end-employer of medical staff are if they get guaranteed a certain baseline level of care that isn't "go to A&E and pray they fix you up".
That is just question of definition of 'socialized medicine' term. Many universal healthcare systems in Europe are based on public/government insurance, and mixed private and public healthcare providers.
You qualify for it and then pick a company to administer benefits. Some companies have great benefits for elders, some have great benefits for young people and families with children.
Socialized medicine does not require single payer, for instance look at how Australia manages their system. Single payer is the exception, not the rule.
