I think at the end of the day reactionaries don't really care about truth, logic, or reason; all of their talking points are just propaganda designed to further their agenda. When right wing pundits complain about schools being Marxist indoctrination centers it doesn't really matter if schools teach Marxism or not, I think they just want to de-legitimize the school system because even though it is anti-communist there are aspects of a liberal arts education that contradict the reactionary world view.

For example, I think for reactionaries even a white-washed version of history that teaches about Native genocide, slavery, the white reaction to the civil rights movement etc. but downplays the worst aspects and makes excuses for it is too much, they would rather that students just not learn about that stuff at all because there is a chance that students might question the version they were taught or decide to look into it further and discover the truth. So by attacking schools as "Marxist" they can pressure sympathetic school boards to rewrite the curriculum, convince parents to homeschool their kids, and so on and prevent more students from being taught anything that could threaten their worldview.

And to add to that, it seems like the ruling class in general wants to move away from the liberal arts model of education and take stuff like the arts, humanities, social sciences, etc. out of schools in order to transition to an education system that focuses solely on skills and employment, so de-legitimizing the current education system helps to further that agenda also.