I am a lawyer.
No, you don't need to admit wrong to be fired for cause. You don't even need an actual judgment against you. If either were true were true, hardly anyone would ever be fired for cause.
No, FU does not need to get brought into the lawsuit and pay money in order to claim a termination for cause. Their name is already all over this. They will likely claim reputation harm, damage to the brand, conduct against FU standards, etc....
Ultimately, this will all depend on what's in the contract (which for HCs typically have morals clauses that cover things like committing fraud in the scope of one's job duties).
But remember, you are talking about the same school that fired the shark ****** for cause because he allegedly lied about receiving death threats at a press conference. Assuming Gapier has another terrible season, why wouldn't they fire him for cause for lying to a recruit about benefits provided as an enticement to commit?