It's the woman's decision anyway, so from a logic point of view, it makes the most sense that women ask men out, instead of men having to bang their heads against 1,000 different walls until one opens up. Women's ratio would be significantly higher.
But, yes, due to fundamental instincts, the men have to do the initiation. And paying for dates. It's not a social construct; it's fundamental to our animal natures.
Because women want to be led by a man who can take charge, even if it's only an illusion, so women will never do the initiation [unless they're trying to prove women can initiate--or, unless they are trying to get with a guy way out of their league].