I'll defer to those who are more versed in the history of political theories but I'm not aware of any arguments against taxpayer-funded free public education that were anchored in an argument against "socialism" itself. This isn't to say there wasn't pushback but in terms of historical "public ed is bad because socialism is bad" arguments, I wasn't able to put my hands on any before the modern era.

I would offer, though, that your body text speaks to the answer to your question. If we think of socialism in the most general sense, that is, setting up systems that benefit every member of the community regardless of their income, there were popular movements against public education from the beginning. In many cases, the pushback came from wealthy white men who did not see the benefit of paying for other men's sons' education. Some of the earliest arguments for taxpayer-funded education by men like Benjamin Rush and Thomas Jefferson were crafted to help persuade those men that it would be a net good to fund education for other men's sons (and daughters.) Their arguments were mostly successful as most new states that joined the union included education in one form or another in their state constitution.

As you noticed, a concurrent wave of pushback around "other people's children" was around race. White Southern men made sure free and enslaved children wouldn't benefit from any form of public education by outlawing literacy for Black children (and adults.) In Northern states, it varied from state to state, city to city. As an example, communities with large Quaker populations were more likely to have Black and white children learning together, the school funded through collective donations and tuitions. Meanwhile, NYC set up a complicated system of funding that was a combination or public and private dollars and members of the committee that distributed funds went back and forth around who should get funding. As you identified, race and religion were two variables that informed those decisions. (This answer to a a previous question on religion in American schools may be relevant to your interests.)

The 19th century saw a sea change around the goal and purpose of public education. Although early advocates wrote and spoke to the role of schools in supporting the development of informed citizens and good Americans, this sentiment became so entrenched the entire nature of the school curriculum shifted by the end of the 19th century. This lead to a shift in the curriculum students experienced from a classical focus (Greek, Latin, some math, some sciences) to a more liberal arts or modern one. The idea behind "common" schools was that education shouldn't be something only available to some men's sons, but to all men's sons (and daughters), especially those who had newly arrived in America. (More on that here.) Again, I'm a bit out of depth in terms of political theory, but my sense is that there is an argument to be made that public school advocates actually leaned into a socialist argument. The pushback against common schools for non-disabled white children of all classes was minimal, but the focus on race and religion endured.