This theoretically already happened during WWII, though...
Uh... did the US send out female soldiers? I must confess I've never heard of it.
Sort of.
WWI and WWII are basically the reason why women are allowed to vote and hold the full range of jobs in the US.
Also, @Nikov: I may be totally wrong, here, but I do have a female relative who was in the Hitler Youth--and from what I've heard about the society of the time, the women were busy doing men's civilian work at home, and dislodging them for the front lines would have been stupid due to killing the support base. Their job in the ideal society was to make Aryan babies and do housework--and that was basically it.
The evacuation west really didn't work that well. All I heard was that bombs kept on bombing the shit out of the little farming village in question, and when the Red Army rolled by the only reason why my relative and her family weren't raped with the rest of their community was that they had a prisoner to vouch for them. Hell, she didn't even know why they were fighting, or who the opposing army was, or anything about the Holocaust. So... I don't think that the operation and decisions were quite as coherent as you described (based, of course, on solely anecdotal evidence).
I sort of feel like the vast incompetence of Nazi Germany in general should not lead us to assume that they had the correct, or incorrect, idea about women in infantry units. I don't have the right answer, either, but I don't think we can say their reactions to a perceived need were necessarily rational or good.