And if you’re actually kind, fair and decent to women you will have people who rebut the rumors.This is very naïve. For this to work, either people would have to be omniscient, or some karmic mechanism is ensuring that "justice always prevails." Let me assure you that neither is the case. I know this, because being different and being a minority, in various times and places, was enough pretext to let people attach falsehoods to you, and have it widely believed. We know this from false accusations in the Jim Crow US south. I know such things from my personal experience.
However, those mechanisms aren't the only ones. No-one is completely immune from such accusations, except for fleeting periods of extreme popularity and societal goodwill. A lie will get seven times around the world, before the truth laces its boots. This, too, I know from personal experience.
The question is this: Do we want mob mentality to be the arbiter of justice? Nearly a millennium of jurisprudence would firmly tell us: NO!
What's more, the mob mentality is clearly sexist! And it's the mob's sexism which is the root of the problem. On average, isn't there a much stronger mob reaction from a woman's accusation of sexism over a man's? It's this difference that gender-skews the cost/benefit calculation. This difference is itself sexist.
Justice doesn't come reliably from the mob. Instead, what we get is bias that results in more sexism. Funny that.