They have a board, yes. They don't have to get public approval of any move, hire, fire, promotion, etc. Neither does Alabama. (The last time UA went a similar route was with DuBose and we see what the public knows about coaching hires.)
Moreover, and back to the point of UCLA, why should Cal have any say so on moves UCLA makes? Better yet, why would UCLA want any input from anyone from Berkley? (Cal-Berkley is one of nine other schools in that system so their opinion is likely moot.)