Another aspect of this issue is that men play the “dad” card dating to seem more appealing whereas women often have to downplay our role as mothers for the same reason. There’s a massive double standard, which is why you see men with kids in dating profiles —even if they don’t have any.

I agree with your wife. I don’t think it’s as much as judging their ability to be good parents as much as recognizing that some of these “single dads” who take every opportunity to mention it aren’t always all that active in their children’s lives. Understanding how involved they are (regardless of the custody agreement) is important when we’re looking for a partner.