CAPT Crozier's boss (Admiral in command of the Carrier Group) was literally down the hall - and he was given the personal cell phone number for the Secretary of the Navy. He apparently ignored both those channels and sent out this letter in the least secure way possible. A letter that provided no indication of the effectiveness of the steps being taken already and was basically meant to be leaked to the press. This wasn't a case of where he was being stonewalled and this letter was a cry for help. He was getting help, people above him were asking what else he needed, and instead of giving them any time to actually get stuff done he decided to throw them under the bus because he was worried if he brought up the concerns he voiced in his letter that he would be told to be quiet.
I'm reminded of the scene in Raising Arizona, in which some bank robbers rush into the bank and scream, "Everyone freeze! Get down on the floor!" and an old codger says, "Which is it you want, young fella, 'cause if'n I freeze I can't get on the floor and if'n I get on the floor I got to move?" *
You can hardly complain that Crozier sent the letter "in the least secure way possible" and ALSO complain that it did not contain detailed explanations of the situation.
So, which is it you want?
So the Captain did what he knew needed to be done -- he copied enough people ON A MILITARY NETWORK that he knew it couldn't be swept under a rug and hidden. Enough people would know there was a problem that some useful action would need to be taken so the ship's crew could be saved and (hopefully) a new, healthy crew could be made ready. The fact that someone leaked it to the press probably means that no one who received it was willing to make waves with their name attached. But that's on them, not the Captain.
The question remains, where was the Admiral in all this? I would like to believe that they flipped a coin to see who would throw themselves on their sword and report this problem. But that's unlikely. It's far more likely that the Admiral didn't want to make waves.
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* Or words to that effect...