You have to try to see things from their perspective. Any option they could choose is going to be bad, so they're incentivized to choose what they perceive to be the least negative one. Sexual/gender groups tend to be hyper active on social media and organization - they are also extremely vigilant in trying to socially damage any perceived enemies. Had Stack Exchange chosen to not act they would likely be facing an angry mob of people who are hyper active on social media hounding them about not being inclusive, engaging in [--]phobia, bigotry, etc.

They probably simply felt that firing one moderator was easier than dealing with what was likely to become a deluge of public relations damage if they did nothing. Depending upon the laws in New York, there would even be the potential of them facing discrimination or other lawsuits. Regardless of the merit of the claims, it would be enough to severely hurt the company. For instance see Ellen Pao vs Kleiner Perkins. Even though she decisively lost every single claim, their company's name and image was dragged through the mud for months in many major media outlets that draw tech oriented readership. And that's a company that doesn't, in the same way, rely on general populace brand value.

The solution, like many things, is to never put yourself in this position to begin with. And that starts with paying for your labor.

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As for the mod selection process itself, it's the same thing everywhere. In general moderators are hired from the community. The problem is the people you don't want to be mods are the ones that'd be most interested in such a position, whereas the people you do want are generally not going to be willing to do it without compensation. There are some exceptions of course, but I think it's becoming increasingly clear that they are indeed the exception.