The world is full of countries with more equally distributed healthcare systems than the US with better public/private balance. The one that gets brought up most often here is the UK NHS, but there are plenty of others.
(Ironically, even if you limit yourself to actual communism, Cuba has quite an effective healthcare system. It just relies on not having freedom of movement so doctors can't move to better paid countries)
"The infant mortality rate in Cuba is lower than it is in the United States and is among the lowest in the world"
Not perfect, but far better than you might expect.
How many countries with socialist health care systems then produce new drugs for patients? I can tell you, it's about none. Most of the new drugs worldwide come from the US market. There's no way going around that fact.
There's also the question of what kind of research is prioritised by the profit motive.
(Human Genome Project is probably the flagship example of such international public research)
Novo -- focusing on diabetes; about 15,000 employees in Denmark: https://en.wikipedia.org/wiki/Novo_Nordisk
Novozymes -- enzymes; about 6,500 employees in Denmark: https://en.wikipedia.org/wiki/Novozymes
Coloplast -- colostomy bags etc. https://en.wikipedia.org/wiki/Coloplast
Genmab -- cancer etc: https://en.wikipedia.org/wiki/Genmab
These are 4 directly medical companies that also are all 4 part of the OMX 20 stock market index (similar to NASDAQ 100, the most 20 valuable public companies). That index also has 2 hearing aid companies (GN and William Demant)
One source estimates 44.000 employees in biotech/medtech in the Copenhagen/Sweden "Medicon Valley" area.