Everyone considers Canada's system to be socialized medicine, yet the vast majority of doctors are in private practice, and the majority of hospitals are not government-owned.
No, they do NOT.
---You just nailed the difference when you said
"the vast majority of doctors are in private practice, and the majority of hospitals are not government-owned" and that's BECAUSE Canada's system is
not socialized,
it is single payer.
*The National Health Service in the UK
*The USA Veterans Administration healthcare system♣
Those are examples of socialized medicine.
The US Government builds, owns and operates their VA healthcare system and everyone working at VA is a government employee.
The United Kingdom, same thing....
and the story with the UK was simple practicality because in 1948 much of the former "patchwork" British system had been bombed out and reduced to rubble, and most British subjects were unemployed and broke, and had been through some pretty awful parts of the war.
King George decided that his subjects had been through enough and that Clement Atlee, a Labour MP, would set it up.
Canada and the USA both have had fully built out mature systems, so there is no need to reinvent anything.
The problem is NOT quality or availability of care, the problem is all the competing private insurers all grappling for their piece of the pie so they can show profit to shareholders. The reason single payer saves money is because it eliminates all of that overhead, Uncle Sam is the PAYOR.
We pay taxes, Uncle Sam pays the healthcare system.
Nothing about it is socialized except the insurance.
Medicare A & B is single payer, although in the last twenty-five years or so we've adding ideas like Part D, an optional, federally-regulated, private insurance plan that helps pay for prescription drugs.
Thankfully as spouse/caregiver to a 100% service connected disabled veteran I get to use CHAMPVA instead of Part D, for drug coverage and as a gap policy to cover the 20 percent Medicare hole.
What a lot of people are proposing is utilizing a kind of adaptation to a Medicare For All setup.
♣ The VA healthcare system differs from the British NHS in that some veterans must pay a reduced subsidized monthly premium for their VA care if their conditions are not service-connected and their income exceeds certain limits and they are not sufficiently disabled.
VA uses a Priority Group system which is explained here.