Frankly, coming from a London perspective, if a man is giving up his seat for a woman just because she is a woman it is normally seen as quite creepy (even if not intended that way). Because people are so used to men and women being equal in the seating stakes giving up a seat is like doing the woman someone a favour, and for understandable reasons the implied favour can be uncomfortable/distressing to the woman. Nearly all the women I know have come across situations where a man thinks that being kind to a woman deserves something back like a bit of conversation, maybe a phone number, maybe a picture etc. So do not think it's looked down upon just because women want to be seen as equals.
For many areas this is a good question and certainly there are a lot of areas where the train provision could be improved greatly (where increasing train length is a fairly cost effective way of increasing provision.)
However London is a different beast. There are many public transport links in London where it is physically impossible to provide enough provision to allow everyone to sit. Even in non peak periods, with trains running every 2mins, it is common to see large numbers of standing passengers on many of the central tube lines. And then in peak, it isn't whether you get a seat or not, it's whether you manage to fit on the 2nd train to arrive at the platform or the 3rd. London is transport is simply at such high capacity that it is preferred to get more people on a train than to leave some waiting just so that a relatively small subset are lucky enough to get those extra seats.
I asked a few women in our London and Manchester office - I was curious about this. I'm in my early 50s and was brought up to offer my seat to a woman - my grandpa giving me a good clip when I wasn't respectful enough and as a miner he had a mean clip.
The women in the north would all be pleased to be asked but most would politely decline, although a couple of my age would accept if a seat was offered by a much younger man. The younger ones would assume a middle aged man was being a gentleman and a younger one was hitting on her but all would politely decline and not find it creepy
The women in London of my age were similar, none would find it creepy if I offered and most would decline unless they were especially tired or footsore. The younger women were surprised it was a thing and wouldn't expect it.
All were amused that I read a train forum though so I guess that's one thing.
I will go on offering my seat, I've been doing it for far too long to break the habit easily. If I find that I am causing offence I might make an effort to stop.
I most certainly will not respond to rudeness though. I have climbed over people to get to a window seat before - I probably shouldn't but thats just rude - and I'm happy to ask people to move their bags. I'm sure its just me but bags on seats seem to be more of an issue in the north than the south - apart from airport routes.
I won't offer my seat to a child as a matter of course - but a tired mum and a tired young chlld both needing a rest - then I'm happy to normally. If they demand then they get to stand though.