You should act as an intermediary or have a teacher friend reach out to the family. This may be a misunderstanding. Kids take things out of context all of the time. If things are smoothed over with the family, this will probably resolve the situation. Even as an attorney, I have found that the best way to resolve these things is to go directly to the complainant and having a conversation.
I do not recommend getting lawyers involved. This will blow over regardless.
The main problem is the besmirching of her reputation via Facebook. She is the most requested substitute and has never been in trouble before. She is taking this hard.
It doesn’t have to be the student who caused trouble. It could be she just reported the news and her mother (possibly this man’s daughter) flew off the handle.
OTOH, even if the mother was complaining on FB, doesn’t mean she was demanding teacher’s head. Might be just the publicity generated on FB scared the principal.
BTW, you are totally right about people working things out directly. Best to man up and face the issue, rather than sending in someone else to slap at them. Even the Bible says this.