Is there evidence of that?
There are formal and proper reporting channels for things like this. He went outside them, on purpose, to ensure the letter got leaked.
Not everyone agrees this was the right call:
https://armedservices.house.gov/press-releases?ID=2EFC2A13-1...
The other has been widely reported, this CNN article has quotes from SECNAV.
https://www.cnn.com/2020/04/02/politics/uss-roosevelt-comman...
When the captain made the decision to send the letter, wasn't he at that time superior to you in rank? But it's no problem for you to second guess his actions?
I guess I don't understand this chain of command business very well.
The Captain sent the letter to many people not in his chain of command, including someone who sent it to the newspaper (as he most likely expected would happen). That's not done in the Navy. Sending a letter like that to his superiors was perfectly acceptable. Getting it on the front cover of Navy Enquirer is not.
Yes, you are correct. You don't under this chain of command business at all.