Outrage mobs don't need a legal setting to ruin someone's life (or livelihood).Question: Would we, on average, expect an outrage mob response of the same size and magnitude when a man makes such an accusation? Whether or not this is justified by historical injustice is irrelevant here. What's salient is whether or not there is a gender skew.
If there is such a systematic and large societal gender skew, then we should expect people's cost/benefit calculations regarding the exposure to the risk of such accusations to also be skewed in a way that is large, systemic, and gender unequal. In a word, the way our society works around accusations, current day in 2021, is itself highly sexist.
Therefore, if we don't want systematized sexism, then we have to eliminate gendered skew in these cost/benefit calculations. We already know the mechanisms for the way out of this. It's codified in various legal systems, and in the values of historical liberal societies and philosophies. They are called respect for evidence, innocent until proven guilty, and due process. When society applies these principles gender neutrally, the gendered skew in individual cost/benefit calculations will even out, on average. Society will have eliminated another form of sexism, and the world will be a better place.
When one says "believe women" somehow in preference to believing men, this is a contributing factor. To avoid the gendered skew, it would be obviously impractical to say, "believe everyone." Hence: respect for evidence, innocent until proven guilty, and due process. Applied gender-neutrally, this is our way out.
In short, the tremendous power we've given mobs based on accusations not-requiring evidence is itself highly sexist, and this distorts our society to also be more sexist.