Would testimony (e.g., a statement under oath) constitutes proof under Russian law? It is, under New York law. Of course, to win on a claim, one would have to meet a certain burden of proof, based on assessments of credibility and an evaluation of competing evidence (including testimony of other witnesses). So when you mention a need to "back it up with evidence," are you saying her word is not enough, or is not enough to satisfy a burden of proof to win or defeat a claim in court?
Are you suggesting that anyone who walks into a police station and makes an accusation has their word accepted at face value with no corroborating evidence, that the prosecutors automatically will accept it -- estimated 99% conviction rate in Russia -- and that the accused will sit back and not say, "S/he's lying" or "S/he misunderstood"?
"Backing it up with evidence" might be notes from phone calls, a paper trail, an audit of books and bank accounts, corroborating witnesses, police investigations, etc.
Going to the police doesn't even necessarily mean testifying: it could be providing enough for them to decide to investigate and gather their own evidence.
If he were to bring a claim, wouldn't he have the burden of proof, or is that not the case in Russia? Also, in New York, I believe he would have to show special damages, or a specific harm. He is a public figure involved in labor negotiations about conditions of workers. I doubt he could establish any financial suffering from her saying she was not treated well in her job. Nor could he establish malice, which is necessary here. Of course, I have no idea about the laws there.
So far, no Russian legal experts have either discussed this in the media or volunteered to post about the specifics. There are much more stringent libel laws in Great Britain, for example, than in NY. There are different burdens of proof depending on place. Without knowledge of Russian law, any answer would be speculation.
In any event, my point about threats is, what would be her incentive for bringing a civil claim or reporting to law enforcement for them to decide whether to bring a criminal action? In a hostile environment, which is described on these boards as, to some degree, lawless, contemptuous of a system of legal justice, and governed by politics and thuggery (as also evidenced by the attack on Filin), she would be subjecting herself to further threats not to testify, which could be physically, emotionally, and financially dangerous to her and her family. What would she achieve by testifying or even by going to law enforcement? Does one really think any positive reform would result (that calls for speculation, but one has good reason to make a pessimistic cost/benefit analysis in this regard.)
There are plenty of reasons for it not to be worth it for her personally to go to the police: that's up to her unless there is the equivalent of a subpoena, where she could be forced to testify, based on her already public accusation.
As far as what she could achieve, justice perhaps, on behalf of the dancers of a venerable institution that was her dream but has disappointed her so. Not letting criminals pursue criminal activity and hurt other people is another reason she might. Of course it's up to her to decide if it's worth it, but, in the meantime, the people she accuses in the press of criminal behavior without pursuing criminal charges through the system have the right to point her to the process and to defend themselves against her accusations in any legal way they see fit.