I believe there is an overly simplistic view of how things work at this level. Everyone assumes that you tell the joint chiefs to "jump" and they say "how high?". Having spent some time around general officers I suspect you have to stay out of some of these affairs. I suspect the decision to remove this captain was made long before it reached the executive office. The executive office may have chosen not to interfere to avoid ruffling feathers. Pretty much, all of the people at this level are powerful, confident and opinionated (inside and outside of the military). Right now you can't afford to alienate them, when you may need to depend on them in a few weeks for who knows what. Its a delicate balance.
I found it sad he was relieved for trying to take care of his men. I wondered how the letter got out. I haven't seen anything definitive on whether he released it or not. If he released it; he had to know he was done. If a 3rd party released it, then thats too bad. My first question to Shelley when I read the article, was does he lose his command? No branch of the military likes to be embarrassed or called out in public.