I’m not Taiwanese, but Chinese American. Yes, everyone is supposed to try to pay, because otherwise you’re just freeloading. But the one with the most “face”/status is supposed to pay, like the family matriarch/patriarch, boss, etc. (Obviously there are exceptions, like if it’s your first paycheck and you’re treating your parents.) Some will even arrange it before dinner/while making the reservation.
So if you’re a “manly man”/someone in charge of your own life, you should figure out a way to pay. Some women care and some don’t. I’m not Taiwanese, so can’t say how many Taiwanese women would care.