It is not in any way bigoted or transphobic to believe that a person being a man or a woman, and therefore the pronouns used to refer to that person, are determined by that person's sex at birth. This was the accepted belief of the vast majority of the human race up until about 20 minutes ago, when it suddenly became a bannable offense.
StackOverflow going all-in in support of social engineering on behalf of political activists is baffling to me. Where does this end? Is the goal really to force all of society to change to placate a small group of activists? A reasonable compromise would have just been to allow "they" to refer to someone in a neutral way, but I understand that this is actually not allowed if the person being referred to has stated they would like "he" or "she". This is what makes me think it's more about social engineering than actually protecting anyone from harm.