It could also be the skating director is trying to get students back. Some coaches think they own their students, but the students have the right to leave at any time. From the OP email there's no indication the new coach did anything wrong. The students could have informed the skating director and left for the new coach.
According to PSA guidelines Skating Directors/Program Directors need to be mindful that they do not make decisions that
unfairly benefit themselves, other close associates or family members over other staff members
So if student's want to leave the director, they can. The director cannot use their position to benefit herself by pursuing charges against the new coach.
The Director should provide the coaching staff an atmosphere to resolve any differences they may
have on their own. If differences cannot be resolved, then the director may need to step in and
help the coaches find some common ground on a particular issue.
Since there's no 'differences' between you and the two coaches, the director has no role and should butt out.
The entire 'soliciting' document.If you haven't read it, you should. Soliciting is fairly clear. In the OP case it doesn't appear that any soliciting happened. What happened to other students isn't the OP's problem.
FigureSpins approach is correct. Send a polite email to the director. If she pressures you, you can always use quotes from the documents above to clarify your case.
My only suggestion is when other parents try to drag you into the conversation that you avoid the gossip and make it clear you have no interest in getting involved.