I think women's clothing had a little to do with it. Huge skirts made it impossible for ladies to open a door by themselves, and make it through before the door slammed on a petticoat. At one time, dresses had huge side panels, [think France and the court of Louis and Marie] the lady had to turn sideways to even get through a door!
I can't quote you a site, but consider some of our other customs that were originally a safety concerns. We shake right hands with a stranger, in modern times, just a greeting . In previous times, the right hand was the hand probably would have held a weapon, it was a check to see if you were about to die thing.
Consider weddings, for what purpose do brides and grooms in modern times dress up a bunch of friends and family and call them the wedding party? It is now just a custom. But in previous times, when marriages were often political events, not romantic, it was a safety thing to show up at your wedding with groomsmen, it might be a trap to see you dead, and to take your estate away from you while you are gone to get your bride. Brides, too had wedding party, so she could not be kidnapped and held for ransom, or "ruined" so she would be less of a prize, or no prize, to another man. "Ruined" bride = her father could not use her as a pawn in any political scheme. And then there was that thing about dressing up a bunch of women to look like you to fool the devil, in case he wanted to do mischief concerning the bride.
So your person might well be right, it might have been hiding behind a woman's skirts, quite literally, to allow a woman to enter a door first. .