“It absolutely chills free speech”
I don't think that this angle is going to be successful.
States do have the right to set the content of their curriculum.
They can't stop the teacher from saying what he wants outside the school, but in the context of the education system, they do get to decide what goes.
The Scopes trial dealt with this point:
https://en.wikipedia.org/wiki/Scopes_trial
The Scopes trial, formally The State of Tennessee v. John Thomas Scopes, and commonly referred to as the Scopes Monkey Trial, was an American legal case from July 10 to July 21, 1925, in which a high school teacher, John T. Scopes, was accused of violating Tennessee's Butler Act, which had made it illegal for teachers to teach human evolution in any state-funded school.
Second, the lawyers argued that the statute violated Scopes' constitutional right to free speech because it prohibited him from teaching evolution. The court rejected this argument, holding that the state was permitted to regulate his speech as an employee of the state:
He was an employee of the state of Tennessee or of a municipal agency of the state. He was under contract with the state to work in an institution of the state. He had no right or privilege to serve the state except upon such terms as the state prescribed. His liberty, his privilege, his immunity to teach and proclaim the theory of evolution, elsewhere than in the service of the state, was in no wise touched by this law.