Men experience sexual access to women as a scarce resource. That is why it is predominantly men who do the approaching. For this thought experiment to be meaningful, that situation would have to be reversed. I don't think that is likely to happen, but if it did then, yes, women would be just as awkward as men are now.

Also, in my experience, women do "approach," but indirectly. I could always tell when a woman was interested, even though she did not express her interest explicitly.