In particular, he made a bunch of statements which would make me concerned about him being in any sort of managerial role over women. He doesn't seem to have tried to walk them back or show some sort of change of heart (or "c'mon guys, it was all just a bit"), so I think his almost-coworkers had a point when they started that petition against him.
People's ideas do matter. Ideally misogynists should not be in charge of women.
Everyone has a right to work (irrespective of position), to live freely, own property, etc..
Justifying the opposite because of someone's ideas is a very slippery slope. And the fact that politics is a game of perception makes it all the more dangerous. If the mob rules, the true rulers are the people who control the mob (an even smaller amount of people)
There is a lot of people with stupid ideas in the world. You cannot punish them all just for their ideas.