I have lived in a number of countries during my career and I have personally experienced two overseas socialist healthcare systems in my life, one third world and first world.
In the third world country, I had to have a very niche test that could only be done at the largest hospital in the city, a government owned hospital. It was a terrible experience. The corridors were packed with sick people sitting and waiting on the floor, it was hot and the ventilation was poor. I had to wait for hours to be seen while hoping I would not catch some disease while feeling incredibly sad for those around me that who must endure that hell regularly while for me it was just a once off.
In contrast, in the first world, the hospital care was just as good as the USA but without dealing with a billing department(can't remember why I was there, might have just been to see a physician for a cough). But from what I was told by a doctor and nurse I befriended while I was in the country, everything was done on a government budget. You want an MRI? If you don't have insurance, the expense comes out of the physicians allocated budget. What happens if the doc has expended the budget? You wait until the next budgetary cycle. Want a procedure done (that is deemed non-urgent)? Its free, but you need to wait months until there is space for you. Non-urgent includes cancer that won't kill you ... a friend's mother waited 6 months to get her stomach cancer removed. She was uninsured and it was free. Good deal (the surgery was something like 12 hours long with multiple surgeons working in shifts!) but they knew that if probabilities did not work in her favor, the cancer might have spread before they could remove it. They decided it was not worth selling the house and other assets to pay for private surgery. It was a good call in retrospect, but if it were my mother, I would have sold my house.
Another friend had a benign brain tumor. The doc was very reluctant to send him for an MRI. Three months later he got one. They instantly sent him for brain surgery. Removed it (for free!). But had it been malignant and spread, the time it took for the doctor to decide to use the MRI might have killed him.
The doctor friend I mentioned above, would tell me government bureaucrats would fight him on his spending, analyzing each of his patients and second-guessing his expenses. He hated them with a passion.
I remember only reading complaints about the UK system until ACA began to be debated, then suddenly it was the benchmark of great healthcare.
tldr; socialized medicine is great, and I would definitely support it, but don't fool yourself into thinking it's all that great. The upper-half of the middle class would still take private insurance and would still prefer being seen at private hospitals where they get the best care no expense (or bureaucrat) spared.