Well, that's one option. However, we just don't know what would have happened had the Referee not caught the ball. It's also possible that the defense would have fallen on the ball in the EZ, resulting in a TD. So that's not the choice I would have made.
Football rules provide that a ball which touches (or is
touched by) an official who is completely in the field of play is a live ball, and the touching is ignored. So, the correct decision would be that it is the offense's ball where the QB fell on it, with the down counted. A "rub of the green" play, for you golfers out there.
What many people miss on this play is that there
should have been a flag on the play against the offense for illegal motion, since the QB was moving forward when the ball was snapped. Honestly though, I don't know that I would have thought to catch that on the field.
The root cause of the problem in the play was the non-standard pre-snap positioning of the Referee, almost directly in line with the snapper. The preferred position is outside the tackle (or tight end) on the wide side of the field. It appears that he had his eyes on the QB (one of his pre-snap "keys") and the ball took him by surprise. He should have just let it go to the ground and allow play to continue.
Invoking the inadvertant whistle rule and replaying the down from the previous spot removes any consequences to the offense for their screw-up, which is unfair to the defense.
Incidentally, despite this fiasco, the officiating crew was chosen to work a state final game in Kentucky. Obviously, this was an experienced crew who had a "one-in-a-million" play happen to them. Hopefully, this play didn't have an effect on the outcome of the game.