>In the private sector, teachers get paid much better and have the freedom to actually run their classroom in the way they see fit.

While this might be true wherever you are from, it is definitely not universal. Every US state I have lived in has been the opposite.

At a private school, the advantages to teachers are the student base is self-selected and can be expelled, and the school can teach things a public school cannot (often religious).

At a public school, pay, benefits, and job protections are much better.

Being able to "run your classroom the way they see fit" is a school-by-school work culture thing, but a teacher with more job protections would always have more leeway in how they ran their classroom. A teacher working at-will could never truly run their classroom the way they saw fit, because "do X or you're fired" is always a possible ultimatum.