1. Until relatively recently, frum Jews didn’t date. Restaurants (kosher or otherwise) are a 19th century invention. “Courting” (as it was called in the English language) traditionally was done at home, with watchful parents present. Going to restaurants or other public places is not a custom based on halacha.
2. In western cultures, the man is demonstrating his ability to support the girl in the style to which she is accustomed. Traditionally the boy would probably find a way to show off his Torah learning in front of her family (which would demonstrate ability to earn a living, traditionally). If she is living at home (as was the case traditionally) she would want to demonstrate her skills.
3. If a man is looking for a “modern” girl who believes in “equality”, he expects her to offer to pay for things and would drop her if she doesn’t.