The thing is that when Sarek is about to tell her how Vulcans achieved diplomatic relations with the Klingons, he makes it very clear that it was a special case.

SAREK: Be very careful that your assumptions are not being driven by your past.

BURNHAM: This isn't about what happened, Sarek. It's what's happening now. How did the Vulcans achieve diplomatic relations with the Klingons?

SAREK: That is a solution particular to us. One cannot assume it would work on a ship commanded by humans.

BURNHAM: Sarek, please.

SAREK: I caution you, be considered with how you use this information. You cannot save lives that have already been lost.

It is telling that we never hear Sarek tell Burnham what happened. Burnham relays the story about H’Atoria to Georgiou, but how much of it is filtered through her own biases is unclear.

BURNHAM: 240 years ago, near H'Atoria, a Vulcan ship crossed into Klingon space. The Klingons attacked immediately. They destroyed the vessel. Vulcans don't make the same mistake twice. From then on, until formal relations were established, whenever the Vulcans crossed paths with Klingons, the Vulcans fired first. They said hello in a language the Klingons understood. Violence brought respect. Respect brought peace. Captain, we have to give the Klingons a Vulcan hello.

But Burnham isn’t being logical here - she’s ignoring a vital factor which Sarek had already pointed out to her:

BURNHAM: We're facing off. No fire has been exchanged. But they didn't turn on their beacon until Starfleet was on its way to our position. I think they're summoning more of their kind.

SAREK: An odd proposition given that their Empire has been in disarray for generations. But you are describing something out of the ordinary. Restraint in the face of conflict. When a civilisation acts in opposition to its instincts, it may be under the influence of something, or someone, new. Great unifiers are few and far between, but they do come. Often such leaders will need a profound cause for their followers to rally around.

BURNHAM: A war.

Burnham knew. She is brought to the same conclusion that Sarek has come to, that whoever is in charge might want a war.

And yet she insists on shooting first, in opposition to Sarek telling her the H’Atoria Incident and what followed was very different.

The initial contact between Klingons and Vulcans was a meeting of equals and based on a misunderstanding of relations with no larger agenda of war behind it. That is why the Vulcan Hello worked then. There’s no reason to think it might work in the same way now.

In this light, Burnham’s justification on firing first rings very hollow, and is likely more driven by the past trauma of her parents’ death. It wouldn’t have worked, and she was aware that there was a real possibility it wouldn’t work but took the risk anyway. That’s the legal definition of recklessness, and played right into T’Kuvma’s hands.

Inaction may not have prevented the war, but it would have made her less complicit in its beginnings.