I know nothing about early 20th Century dating, Alyson, contrary to what some people might imply (stop laughing, Bob). But I'm sure they mostly didn't 'date' as we'd understand it. Girls were chaperoned, often by maiden aunts or children, and young men had to ask permission of parents to marry their daughters - well, in the middle / upper classes at least. At balls, young ladies had dance cards, on which they wrote the names of the partners who'd requested a dance - it was all very formal. So, I doubt that 'pick-up' lines came into it much.
Men use pick-up lines in situations where they don't know the women, but in society they did usually know each other, or at least know a great deal about each other's circumstances (wealth, position etc.). And, even in the working classes, Edwardians usually courted people they knew. And, I suspect, this was the general rule until surprisingly late - maybe the 1950s. Of course, there were always exceptions.
I'm guessing here to some extent, but one of my grandmothers once made an interesting remark to me. She told me she'd married my grandfather 'to get rid of him'. She didn't mean it harshly, as she grew fond of him and respected him. But she was under such pressure to marry him from her brothers (his friends) and her own mother (his landlady), that it seemed the simplest solution to stop the pestering. That would have been in about 1907. They weren't 'society' people of course.