It is like the Klingon war is her personal story.
That's not a "personal story", that's called having empathy... When you can imagine that folks may go through the same horrible things that you did and you wish deeply to change that. Empathy.
Meaning to or not she drew first blood.
Well, no. She intended to. The Klingons fired the first shots. Even if she was the one to do so, she had a good reason. She was right about the Klingons not wanting to negotiate. T'Kuvma intended to have a War against the Federation regardless.
Really it is not an etiquette thing. It is military law.
Because something is a written Law it doesn't mean that something is necessarily good. The Chain of Command is a highly idealistic thing. It looks flawless on theory but in reality it can get on the way of doing the right thing.
Example: Wasn't George Washington a member of the British Provincial Militia before the Revolution ? What if he followed the Chain of Command and refused to align himself with enemies of the Crown ? He broke (British) Law to do something he believed was right. That's a very Human thing to do. It's realistic too.
If you want to go into a more in-universe example, the Prime Directive is one. Highly idealistic. Is it a good thing to let Millions die because they aren't advanced enough to be aware of you ? A lot of folks would say no.
When the Federation is expecting to lead by example it deserves to have its Starfleet officers do the same.
What do you mean by leading ? Leading what exactly, the galaxy ? Because that's not a very Federation thing. The Federation wants to explore, not to lead. They are hospitable for those interested in joining their little club but that's it. When they encounter a species that doesn't give a damn about their values, like the Klingons, they
try to leave that species alone and not go on a holy crusade to convert them culturally.
That's what Klingons like T'Kuvma think the Federation is trying to do. They are wrong, obviously. There's no massive conspiracy to come after the Klingons "atom by atom".
We have to keep in mind that while the Federation is inspired by a lot of Modern Day Institutions, the UFP is not necessarily a mirror of them. I believe Starfleet officers have a lot more space to object things that go against their conscience than 21st Century Military officers do. We're talking about a Post-Scarcity Space Utopia. I wouldn't doubt Michael gets a full pardon by the end of the Series, so the writers can make a point that Future Humans are better than Modern Day Humans after all.
Discovery has given us not an exploration theme but a war one. Not heroes or those who at least try to walk the walk, but these other types of characters.
DS9 was about War and it was very Trek-ish, Star Trek doesn't need to be necessarily about exploration. Burnham may not be a hero yet. She could be if she manages to stop the war. We have to wait and see. And Lorca was never intended to be an anti-hero IMO. He's a straight out villain. There's no good in him.