There are 2 very different things being called socialism these days:
Social Democracy: This is a country with strong social programs but an underlying capitalist system. Sweden for example.
Democratic socialism: this is *real* socialism where there is collective ownership of the means of production. As far as I know this has always turned into authoritarian socialism / communism with horrible results.
Ironically, even though people (usually Americans) want to call the NHS socialized medicine, the NHS does not produce its drugs or medical hardware. They buy them from private companies.
And it's the same with public education in the US. Educational books and other supplies are sourced from private companies. What is wrong with having a teacher being a public employee?
Even with "privatised" education, also known as charter schools, teachers are still being paid with taxpayer funds.