Religious teachings are protected under the 1st Amendment. Religious PRACTICES are a different matter, but Religious Teaching is prohibited.
Public schools that actively contradict what God considers to be ordained marriage in conflict with what Scripture teaches, or what God-sanctioned sexuality is, are, indeed, moving into the religious thicket. The Pride Flag IS a political statement and a statement that, unavoidably, contradicts teachings of all three (3) major religions on marriage.
Students wearing Pride Flag shirts, is protected speech, but so is a kid wearing a shirt saying "Jesus Saves". What is prohibited is messaging from school figures that coerces students to affirm the Pride Flag and what it stands for. To say "We don't discriminate on the basis of sexual orientation here!" is one thing, and that is a statement of policy that is protected. But the Pride Flag is more than a statement of non-discrimination; it's a statement of affirming sexual practices that are not authorized by Scripture. And NO person should ever be coerced to affirm "Pride", in whatever manner they are being coerced to affirm it in. And no child should be so coerced; the Rightness or Wrongness of this matter is a matter for parents, and not for teachers. If teachers can't accept that, they need to find a new job, or be fired if they can't stay within this boundary.
Students have a right to their beliefs. They do not have a right to harass other students.
It is not inappropriate for a teacher to have a Pride flag in the classroom. It serves as a message to LGBT+ students that they will always be accepted in the classroom by that teacher. Students have a right to feel safe in their classrooms, and teachers have a right to signal to their students that they will help foster that safe environment.
And the presence of that flag is not coercing religious students to affirm beliefs that contradict their faith. That is a ridiculous assertion to make.
Is a presence of a cross just a message to Christian students that they will be accepted? Or would it be reasonable to think that a cross is promoting a certain ideology?