I'm sorry you have to endure such an antagonistic social environment. Where I live (midwest US), it's considered polite for all people hold the door open for all others, regardless of gender.
I'm a male and if I worked in a place where women sneered at me for holding the door for them, I guess I would start only holding the door for men.
Holding the door open for all genders is widely seen as polite and fine, for whoever arrives at the door first, man or woman. It's an optional courtesy.
Having a man "race over" to hold a door, and doing so because they're a woman where he wouldn't for another man, in a professional context, is creepy and weird.
Do you see the difference? What you're describing in the midwest is fine. But it's not what the parent was responding to. The "antagonistic social environment" with "women sneering" you're describing is a total straw-man of your own imagination.
I'm a male and if I worked in a place where women sneered at me for holding the door for them, I guess I would start only holding the door for men.