Setting aside the fact that polygraphs are pseudoscience mumbo jumbo that don't work in any meaningful capacity, and the results of which are really just the vibes of the person running it (with all of their bigotry/biases on full display.)

The bigger issue is that there are over thirteen thousand school districts in the US. If each school board is four people on average, that's over fifty thousand people you'd have to do polygraphs for. And that's if all you wanted to do was school boards.

Trying to get all of those people polygraphs would be an absolute logistical nightmare. There aren't that many polygraphers out there.

And we shouldn't be legitimizing polygraphs anyway. They have time and time again been shown to be absolute bunk, and to discriminate against people with issues like anxiety (or really, anyone who gets agitated when you accuse them of something). The only people who can reliably pass polygraphs are sociopaths, which feels like the opposite of what you want to be selecting for here.