Let's be honest here, how many times would you say the Doctor's main companion has been male? I think there's an argument for that just being Jamie. There's Ian yes but it’s hard to disentangle him from Barbara, and there’s a similar issue with Rory (more so given Ian and Barbara were equal companions from the start whereas Rory came and went.) Harry? Well he came in with Sarah Jane already in place so he was an additional companion, and Sarah Jane was still there when he went.
I guess you could say Adric might fit the bill, but even so he was superseded by Nyssa and especially by Tegan once they came into play. There’s Turlough of course, but again Tegan and Nyssa were already in place when he arrived, and though he outlasted them he only outstayed Tegan by a single episode, and even then there was the female presence of Peri knocking around.
I’m not nearly au fait enough with Hartnell to know how Steven and Ben fit into things, so Steven certainly might count alongside Jamie, though surely Ben came as a pair with Polly, much like Ian/Barbara.
Obviously there are male characters who’ve filled a companion role on occasion; The Brig, Mike Yates, Mickey, Jack, Wilf but none of these were the Doctor’s companion in the same way Jo, Leela, Romana, Tegan, Ace, Rose etc. were and from a public perspective the Doctor is a man who travels primarily with a female companion.
Don’t get me wrong I’d love to see another Rory, and certainly another Jamie, but given how strongly people feel about gender balance within tv and films these days and things like the Bechdel test can you imagine how a male Doctor with a single male companion would go down? Of course the easy solution would be a female Doctor with a male companion

(Actually how about a female Doctor with a female companion?!)
As for his comments about women, he’s hardly saying anything that different to what you might hear being expressed by people who want to see more women in positions of authority within politics and business precisely because they are perceived to think differently and bring something different to the table.