Ige didn't pick Tsutsui. Gov and LG run separately. That's why Tsutsui is still LG, even if Abercrombie lost. Same if Kouchi was appointed LG and then ran for the office. He could win for LG even if Ige lost.

So a problem for Tsutsui is that if he ran for LG again, he could hold the office under three different governors. That's just embarrassing.