Lilmod,
You said
I didn’t listen to the shiur on the chabad.org website, but from the quote, it doesn’t sound like it was talking about ladies walking in front of men – it was talking about the general concept of “ladies first”.
So I brought that to show ladies first means walk in front from the inverse relationship n the Torah
Additionally I’m thinking that the reason for the heater is so that people shouldn’t get all upset if a woman is walking in front of him into a bus and cause a scene (something common in Israel)