Players and coaches should know the situation. Umpire has egg on his face and must suck it up.
Agree, better not to verbalize unless batter runs to first when she isn't supposed to.

I'm commenting on the mechanic, not the play, in saying the word "OUT" should not come out of the umpire's mouth unless there is a situation where the retired batter is heading toward 1st on a U3K when not entitled.

Not to drive this into the ground...
The only time I've ever heard "Strike 3 batters out" has been on television show/commercial/movie umpires.

Is it possible that the order of events was:
U3K, batter runs, PU says, "batter out", F2 rolls ball? Then we have the PU giving bad game situation info, putting defense in jeopardy. And we can better debate whether it can be rectified.

If the order of events in the OP is precise, then yes, bad mechanics, you shouldn't say that, AND bad game situation info.

I already did. ;) :D

OK, I will work on my presentation skills. That is what I though I described in the OP, although I now realize I should have included "batter runs" before "Plate umpire, not thinking 2 outs, says "batter out"".

Unfortunately, the rule (and the direction given in clinics) doesn't say anything about "giving bad game situation info". The rule about rectifying situations where we have put a team in jeopardy has to do only with incorrect rule applications or changed calls.

But at some point you have to change the call. He called the batter out by mistake. If he never fixes it the batter is out and the inning is over.

But what you said a little further might be objectionable if I took it like this. Suppose the sequence was.
"Strike three."
Batter starts to run.
"Batters out."
Catcher tosses ball toward circle.
"#@%@, two outs, batter is NOT OUT"
Pitcher runs for the ball, barehands it toward first base and runner is safe on a bang bang play.

versus

"Strike three."
Batter starts to run.
"Batters out."
Catcher tosses ball toward circle.
Pitcher knowing the umpire was wrong runs for the ball, barehands it toward first base and runner is safe on a bang bang play.
Then umpire says, dang, runner was not out.

Whatever the right way to think about those two scenarios, the idea that the first is more clearly in favor of the defense seems absurd to me.

Doesn't this have the potential for a changed call? Isn't that one of the main points of this thread?

yes

Or delayed calls

I think this is the consensus.
Definite mistake by PU to call an out when BR is eligible to run.
Players (e.g., catcher) are responsible to know the rule and make the play anyway.
No need to apply the jeopardy rule, even if ITUJ the defense would have gotten the out with a throw to 1st.
BR remains safe at 1st.

That's my ruling.

Agree.

Another point for consideration: Unless you're regularly working 12U rec ball, you will see that the overwhelming majority of catchers worth their salt will know to immediately come up with the throw after failing to catch the third strike with two outs. They won't be affected by a plate umpire erroneously calling the batter out. If one does, shame on her.