Two things: a) does anyone else get a Mighty Whitey vibe from the implication that Western medicine may be the key to eradicating the Redface Pox?
And b) if they get to Shogun Ienari, how the hell are they going to do the gender flip? The real Ienari was said to have fathered 75 children, how are they going to reconcile that with a female shogun? They're either going to have to severely reduce the number of children down to the direct heirs and key players or make Ienari the first male shogun in decades. (Yes, I have been reading about the shoguns on The Other Wiki, why do you ask?)
I may have figured this out. Of course Ienari will be the first male shogun in decades. The Redface Pox will be eradicated by then, if one WMG is correct.
Why do they still need a shogun? It's repeatedly noted that with men in short supply, wars are the last thing on anyone's mind, and with it standing armies.
The Emperor had been sidelined almost entirely since the late 1100s. It is not quite at the level of "Why bother with a king, the pope is still there." but it is close.
Actually, that brings up another question: is the Emperor another case of She Is the King? I can't imagine that they managed an unbroken line of male succession, especially if they keep marrying off their princes to the shoguns.