Not necessary. In fact, this is quite rare. About one in ten, by my outdated guesstimate. It's usually not worth the girl's time also. Ask yourself, do you want to spend 2 to 3 hours outside the house, with someone you don't really like, just to mooch a $40 meal? The ROI is pretty low. Only the most cheapskate of women will do this. Usually if the woman is making a foodie call, the bill will be a lot higher then $40/pax, to make it worth their effort.
If a girl expects a guy to pay, she could honestly subscribe to traditional values and traditional gender roles.
Or she could be a fake feminist out hunting for a simp to be her lifelong partner. She's genuine, in the sense that she genuinely hopes to find a life partner out of this. Just that she doesn't want a partner who will be her equal, but one who will be beneath her. Your willingness to pay for the first date is one of the signals she wants, that indicates that you will be pliable, and a possible simp candidate.