A working public health infrastructure is a very different beast from socialized healthcare. What you describe is that we had a good public health system in 1918. Not socialized healthcare.

The difference is that healthcare involves caring for the sick while public health involves steps to address health-threats to the community. Public health includes things like clean water, making restaurants meet sanitation standards, vaccination programs, public education efforts, etc.

As an example of the difference, the SARS epidemic was solved through public health measures (contact tracing and quarantine). Health care efforts were notably ineffective. In fact from a public health perspective the main reason to provide health care to people with SARS rather than just shooting them is that you're going to get better compliance if you give people hope that they have better odds of surviving if they report their symptoms honestly. (Public health officials are very interested in figuring out what steps will improve public compliance.)

Until Medicare, the USA didn't have any form of socialized healthcare (and now only has it in a poor form). However the US public health system in the USA got started in the 19th century.

Incidentally the arguments for a good public health system are far more compelling than for universal healthcare. For one thing, public health by the nature of the beast cannot be delivered privately. For another, public health is far more cost effective than healthcare and therefore is a justifiable investment.

Cheers,
Ben