The phrase comes from an era where women were regarded as being in need of special protection, like children, unable to look after themselves or their own interests.
I think there's a more practical side to this. I suspect this isn't just sexism, but that there used to be a solid biological reason for doing this.
When I was pregnant I got SPD after ignoring the advice to not carry heavy stuff and injurying myself lifting a heavy bag. It's no joke! I was in pain for months.
Back before women's sexual revolution, reliable contraception was hard to get hold of, so women would have been pregnant A LOT. Also women wouldn't have had access to moderm medicine to help recover from damage from childbirth.
Before becoming a mother, I only saw "ladies first" as sexist bollocks.
But after realising what a strain pregnancy puts on your body (if not lasting damage), now I get it. In the old days women of childbearing age would have been pregnant a lot more. Men couldn't very well go around asking everyone "are you pregnant - or have you recently had a child or sustained lasting damage from childbirth" could they?!
So, in the old days, maybe, saying "ladies first" could be seen as both reasonable and fair - equity rather than equality.
If I had no access to contraception and was pregnant a lot, I'd certainly appreciate living in a culture that made allowances for that.
I'm not saying that women weren't infantalised, we were of course, but that there may be more to "ladies first" than just that.
BUT anyway - we're living in modern times now with access to decent contraception. It's not reasonable to expect many women are pregnant - or recently pregnant - any more, so it's time we stopped doing this. It's not polite, its patronising!