Or the other way around. Not super relevant for this discussion, but I feel it's important in the understanding of sexism to keep in mind that the discrimination can swing both ways. (In fact, sexism where men are discriminated against often _also_ end up hurting women, which is a phenomenon that holds for many issues related to sexism -- it's in the interest of both genders to get rid of this problem).
It could (and may even happen in small systems), but in reality, women are the ones with significantly less power. But it's also important to remember that not every discrimination is sexism. For example, sexual discrimination designed to correct an already existing imbalance of power is the opposite of sexism (regardless of whether or not you find it just).
It isn't speculation, and why he acted the way he did is completely irrelevant. Sexism is a quality of a (social) situation where women's power is diminished compared to men. It doesn't have to be nefarious or intentional.
> To brand someone as sexist is very damaging to their reputation.
No, it isn't. We're all sexist. To brand someone as misogynistic may be damaging.
> That women in general suffer from a power imbalance is no excuse for attacking a persons reputation with no strong evidence.
No one here was attacked. The moderator did something sexist, as we all sometimes do. It happens. The only way to fight sexism is to point it out. I understand that the moderator must have been embarrassed by being called out, but we should all learn to be less sensitive about such things.