I would argue that public education, medicare/medicaid and social security ARE socialist programs. It's just that we as Americans need to be more comfortable with the fact that socialist programs aren't (in and of themselves) bad. Socialist programs such as these are necessary to check and offset the evils of a capitalist economy.
I'm surprised we've made it this far into the thread without defining the term under dispute, but I would suggest you look up what socialism means.
https://www.investopedia.com/video/play/difference-between-communism-and-socialism/
I think our discussion here mirrors the confusion in the national conversation re: socialism. There is absolutely no precision in the language used. At base, socialism means collective ownership of the means of production and the fruits thereof, whether you describe that collective as "the people" or "the government." Many of the social programs mentioned by you and in that poorly written/reasoned article may have been associated at one point with various political parties that branded themselves as some varietal of "Socialists," but there's a difference between political party affiliation and fundamental philosophy. That some party calling themselves Democratic Socialists, or [fill in the blank] Socialists, supports a program does mean that it is socialism.
Make a concise argument as to why public education is not a socialist program. Because public education wouldn't exist in a purely capitalist society, so how else would you define it?
I think you've created a false dichotomy between "pure capitalism" and "socialism". Again, consider my post about conflating socialism and ideas espoused by professed Socialists. I am currently a Texan, but that doesn't mean all of my ideas are "Texan" ideas.
Here's my concise argument: The decision of The People to fund public education by allocation of certain monies from tax revenues has nothing to do with the state controlling the means of production. In support thereof, I submit to you that (a) most school funding comes from the local level, rather than the federal level, and (b) you are free to send your children to private schools, and in so doing you are in control of distributing the fruits of your labor to that personal end.
