Alas, I do think a lot of this comes down to education, or at least the ack thereof. As you mentioned, hardly any classrooms across American are even teaching CRT at all, so most of your average Americans (not in government or media) who are challenging it are not being exposed to it there. Nor are they reading about it at libraries, or hearing about it from people who are versed in its terms - like yourself. They are being exposed to it through media outlets and conservative personalities. It's here that the caricatures of CRT conservatives create are being spread. This spread is then catalyzed by the fact that many Americans simply do not receive a good education; in particular, lower-income Americans - the ones who would benefit the most from learning about it. It's also this poor education that even makes checking the "budgets and leadership of the average public school system" a very difficult task for some: more generally, the fact-finding skills necessary to question and research state legislation and educational policy.
I'm not saying people with lower incomes are therefore any less intelligent than higher-income folks, that would be offensive and inaccurate. However, they do very often have a lower level of formal education and less access to gaining said education. So, when they are exposed to something like CRT, the critical thinking and background knowledge on this subject simply aren't there. Which loops us back, again, to education. This needs to improve across the board so that people can a) be better equipped to take on CRT, and b) better challenge those who seek to misrepresent it.
This article was a great read, thanks for posting!