I’m partially in agreement here, because it very much depends on the context of the political beliefs, and of those beliefs affect a person’s ability to do that job.

A cop, (a public servant of any kind really), who espouses racist beliefs, on social media, for example. I don’t know if that person should be fired, but I’m definitely going to call into question if they’re capable of properly serving the public, especially if they’re arguing online whether any group of marginalized people should exist or not.