This is correct, under Law 31
31.1 Umpire not to give batter out without an appeal
Neither umpire shall give a batter out, even though he/she may be out under the Laws, unless appealed to by a fielder.
But the dual dismissal scenario is not legal, irrespective of the spirit of the game:
Under Law 31
31.4 Appeal “How’s That?”
An appeal “How’s That?” covers all ways of being out.
and also under Law 20 20.1 Ball is dead
20.1.1.3 (The ball becomes dead when) a batter is dismissed. The ball will be deemed to be dead from the instant of the incident causing the dismissal.
It is not unusual for the striker to be potentially out in multiple way off the one delivery. The one striker could near simultaneously LBW, Bowled, Caught and Stumped off the one ball.
The fielders would appeal, though them laughing to the point of being collectively catatonic is possible. Both umpires would indicate that the striker is out. In the scorebook, under the dismissal order of precedence, the batter is recorded as being out bowled. If the ball did not dislodge a bail, they would be out caught. If the catch was not effected, they would be out LBW. If LBW did not apply they would be out stumped. And yes, the order of precedence does violate the letter of Law 20.1.1.3
There are (rare) instances in a single delivery when either batsman could be dismissed. Most are variations with the non-striker being run out. (To stretch even further the scenario above, after stumping the striker, the keeper then throws down the stumps at the bowlers end with the non-striker out of their ground). The other is “Obstructing the Field”. There was also the potential for the non-striker to be dismissed “Handled the Ball”. These were originally separate laws but since 2017 both modes of dismissal has been folded into the one Law 37 “Obstructing the Field”
Andrew Hilditch (1979, Australia v Pakistan) was the only time in international cricket that the non-striking batsman has been given out for handling the ball.
In a scenario when the striker is struck on the pads, the ball rebounds to a fielder at short leg who throws down the stumps at the bowlers end with the non-striker out of their ground. The appeal is made after the run out effected. The umpire will first consider the LBW, and if they determine so give the striker out and the ball is dead. If they consider the striker to be not out then the ball is still live and non-striker is run out. If the technology is in use, the striker given out but on review it is determined say the delivery pitched outside leg stump and so the decision is overturned then we are in a bit of a mess. I don’t know whether the DRS “resurrects” the dead ball and the rest of the play ie the non-striker is run out stands. I think it should, but I don’t know.
However, in no circumstances are both batters out. Nor can the fielding team determine which one is out. That is the sole prerogative of the umpires under the Laws of the Game.
As a smart alec aside, if that impression is not already ubiquitous , two batsman can be dismissed when only one delivery has been bowled.
The first is dismissed for any valid reason. The second is tardy making their way to the pitch and is dismissed under Law 40 “Timed Out”.