I know lots of people who split. I've always preferred it because too many guys feel like you owe them.
However, girls get a lot of pressure to go out with guys they don't have a lot of immediate physical interest in, whereas guy usually ask out girls they want to sleep with - which provides motivation for some guys to be more over the top about things. Keep it simple like coffee or a beer and chatting.
Society doesn't force men to pay on the first date. It's just well known that that is what happens. Here it's very common to just pay for yourself on the first date. Not too sure where you are from but if I like a girl I don't mind paying at all :) have a nice day :)
Because it's a cultural norm not based on sex but based on proposition. "Girls are asked out by guys" therefore the male is offering, by extension, the invitation. That's like inviting someone to your birthday party and them calling it cultural insensitivity when you didn't buy them a present for showing up to support you.