I may not have dealt with racism very much in <i>Double Jack</i>, but I do deal with sexism.
By sexism, I mean organized and systematized discrimination based on gender. In this era, a woman's gender almost entirely defined her, especially at the upper levels of society. Women were not just believe to be poor voters, they were scientifically shown to be incapable. Men simple could not let this happen because it was dangerous. In the end, determined women won suffrage, but not without considerable agitation and illegal conduct.
Our lead character, Jack, isn't a terrible person or a woman hater, but he does have the cultural beliefs of his time. To him, it utterly inconceivable that a woman could be a wizard because woman is incapable of proper the proper dispassion required of the discipline. This not an opinion to him, but a fact.
And like all people with fixed beliefs, Jack does his best to maintain them. That's just what people do. When faced with a bevy of capable women, he naturally concludes that each one is an outlier.
Of course, our hero changes his mind as he progresses, but his change of mind changes no one else's mind. He may become enlightened, but the world does not. And even if he does become enlightened on one point, will Jack become enlightened on all others? Can his mind ever be fully freed from all the bindings created by his culture? I don't think so, and that's what make an interesting character.