Yeah they should and it's not just because of Last Night. Harada for example had those "SJW" tweets too. It's a issue that deserves to be brought up even when there isn't E3 and Microsoft stage giving it extra spotlight.
And about moderation, can't comment on Twitch mods. But in their forums it was bit weird that I got threatened with a ban when discussing these matters with the outspoken and proud gamergater for calling him obsessed pathetic little man for supporting gamergate and sharing his disgusting views and that he should be ignored. Mod responded with "No need to attack someone based on their personal beliefs. Plus, to be honest, I've never seen an actual gamergator who is misogynistic, homophobic, etc. I'm sure some exist, but people are people and some people are assholes. (and those people SPECIFICALLY will not be welcome) That's all I'll say on the matter, and I strongly recommend the subject be dropped." So yeah, I can see that there's maybe something wrong with mods.