Sure it politics but more so its risk and numbers. Private school teachers don’t have to be back in person. That is a school choice and it benefits a small percentage of the population. Healthcare workers got it first because they have the highest risk, help the most folks, and losing them posed a greater risk of death to the population at large. Public school teachers will get it ahead of private because they affect a larger portion of the population. The idea being if they are safe and able to go back, more students can go back, more parents can be more effective at work, the economy can rebound. Again larger affect for the population at large.
If the vaccine were readily available giving it to everyone would be right, but its not, so tough choices.