Do not conflate politics and grammar, although they are at issue here. Traditionally, the iron grip of "agreement in number" forced this: EVERYBODY HAS HIS OWN OPINION. "Everybody" is singular but gender-neutral, but there is no gender-neutral singular pronoun we can wrest from the available choices, he, she, or it. So we fall back on "they." EVERYBODY HAS THEIR OWN OPINION. To Purists this is an abomination and they insist we should say "he or she," which is inelegant, to say the least. We cannot fix English usage. We can, and should, abandon this old rule: a pronoun and its antecedent MUST agree in number. Why? Because the grammar book says so. I recommend you get a new book, which will have abandoned, if it doesn't already ignore, this "rule." English adapts. We must too.