but why else would the Microsoft board put the guy who wanted to chop it up and sell it to the highest bidder in charge of its performance?
They didn't, not exactly. Elop isn't the head of Xbox, he has been made the head of the Devices and Studios division at MS, which happens to contain the Xbox branch. Elop was put in charge of D&S because MS are going to start making their own phones with the assets they got from the Nokia purchase, and since Elop was the CEO at Nokia and has experience working in that field, he was the obvious candidate for leading that division. The fact that he also now oversees Xbox is tangential to the larger strategy at MS.