How do you know he didn't?
I'm not saying he didn't attempt to work the issues through his chain of command at all, but he didn't follow through. If you go to your supervisor and are ignored, you continue to go up the chain of command (that's why it's called a "chain"). You have the right to go all the way to the top if necessary. Also, you can appeal to the Inspector General if you feel the chain of command is attempting to do a cover up.
No, the article states Provance decided to take it to the press in direct violation of military orders. Constitutional rights to free speech do not give a person the right to violate lawful military orders. Otherwise, what's to prevent a military member from revealing SECRET information to anyone he wants? After all, wouldn't that be free speech?
You give up some rights when you decide to enlist. That's just part of the job, and he should have known better than to violate orders like this.
His "lawyer" states as much.
Horton who is a lawyer and performed legal services for him stated "Furthermore, the command gag order may not have been lawful, Horton said. " And the Sgt himself states, "“My career is over, I thought, ‘Now I can talk’ — almost like it was meant to happen.” . He knew he was not supposed to talk. Service personnel are bound by UCMJ not by the laws of New York, New Jersey, etc. when they are on duty in a foreign land. It is not against the law to report illegal activity to the proper authorities, but he also stated, "If I reported everything I heard, I’d be reporting people every day,” Provance said. “I did not know what was legal. I’m not an interrogator." He disobeyed a lawful order. Had he suspected any breach of law, there are legal channels available for its reporting.