I don't see how that follows at all? Obviously they have interest. As for means, what more do you want? We have a public accusation from a victim in her own words under her own name. Maybe you choose not to believe her, but that's absolutely admissible as evidence. Why doesn't it count here?
The subtext to all of this pontification is clearly not about whether or not private organizations can police their own members. Clearly they can. You don't need to be convicted in court to be fired for stealing soda from your work fridge. This is no different.
What's bugging people in this thread is the same culture war fight we always get into: you tend to disbelieve sexual assault accusations from women, and Lichess tends to believe the accuser. And so sides are drawn up and strawman arguments are prepared. But it's just culture war.
I am not saying that this happens often, there's just no way to know. And maybe it's still better to "play it safe". It's hard, but I've personally grown to accept that I have to live with "being the perpetrator" in the eyes of most. I lost most of my what I considered to be my friends, and my professional career. But the few remaining ones who believe me I now know I can fully trust.