> What specifically did he do recently?
Refused to apologize, even fake apologize, meaning he hasn't changed his views. He also came across as weirdly combative, trying to justify his homophobia by pointing out that homophobia is still the norm in Malaysia or something along those lines. He wasn't smooth about it at all.
> Somehow I don't think it's a good idea for democratic processes to label the side that is wrong as "horrible people."
There are degrees of "horrible people", and in my opinion Brendan Eich does fall on that spectrum, though admittedly he's not even close to the worst of the worst, but he makes the cut. That's my two cents though, take it or leave it. I personally feel pretty strongly about gay rights, so there's that.
> What if the person with the opinion also sincerely believed that his position was not bigoted and was not denying anyone happiness?
I think this is a flawed argument. Bigots almost never seem to believe themselves to be bigots. They think they're saving gay peoples' souls (without ever asking if they want their souls saved) or various things like that. Or they, maybe not overtly, don't really see gay people (or whatever group we're talking about) as real people with feelings.
That's what is so insidious about bigotry, it's not a Lord of the Rings thing where the villain (who conveniently wears black and lives in a scary castle) stands up and says "Muhahaha I'm going to make everyone miserable!". Instead, the villain says "I'm going to help all of you out by forcing you to be just like me, trust me, things will be better this way!".