Lets revisit where chivalry came from:
Women are seen as weaker (lets talk physically). Women were also stuffed in corsettes and fainted all the time. Can someone who can barely breathe open a door?
Some of the old style buildings have extremely heavy doors (where I work they do – anyone is welcome to come to Con Ed headquarters and try the inner set of doors). Since men are generally physically stronger it makes logical sense for him to open the door for her.
We dont pasken like the gemara – we derive halacha from the gemara. Does anyone have a halachic source for a psak on opening doors? It makes a big difference. If it were bottom line halacha, that would mean under no circumstance (including elderly or someone carrying lots of packages) would a man be able to hold the door open and allow her to go through. Since I’ve seen many frum men hold doors open for women, I would say that there must be at least a difference of opinion in how people pasken.
As to what I do: when I reach a door, I hold it open for the person who is behind me.