Outside a very few liberal cities, critical race theory won't be taught in American schools. The word "theory" is kind of the driver here. We don't teach "theory" to small children. There is plenty of time for theory in high school and college.
It's fun when people bandy around "free speech protection" so loosely. As has been mentioned here many, many times with right wing and left wing speech, one might have the right to free speech, but that doesn't shield you from the consequence. An employer, in this case a State, can instruct you what to teach. It's fairly simple. I don't know too many workplaces where unrestricted free speech is allowed. I can have pretty much any political opinion outside work, but I'm not really free to voice and campaign political ideas in the workplace.
Oh, and much to your surprise we're actually taught here in the States that the war was about slavery. There probably were some Southern school districts pushing the "States' Right" myth and other revisionism in schools through the 1970s. Occasionally, you come across some person in daily life or social media who grasped this lesson. You can usually point to the words of Confederate leaders at that time. They were fairly unambiguous for the need to fight for slavery. I personally believe Alexander Stevens the VP of the Confederacy was a traitor to the United States.
The Confederate South was a traditional, agrarian society rooted in slavery. Relative to the United States, It was a small region both in population and economic measurements. Due to Northern industrialization and immigration, the South's importance was getting smaller and smaller unless it could push slavery into Western states. It's been repeated and usually ignored that the slave owning population was very small.