The character of Burnham is quite clear in the show. She makes the decision she makes in the beginning because losing her crew and captain, who is a mother figure to her, was far more important in the moment then standing by the principles of Starfleet and the Federation and doing things the "correct" way as she and others understand it.
I don't think it's all as crystal-clear as you suppose. To me, a lot of it came across as very thematically conflicted and inconsistent.
For instance, in the first confrontation with the Klingons it seems evident that the Admiral is being naive (and more than a little self-righteous) in his attempts to talk (playing right into T'Kuvma's hands, in fact), and his adherence to Starfleet doctrine is just setting himself up for the "surprise" attack that kills him and his crew. Georgiou comes across as only slightly less naive and doctrinaire. Meanwhile Michael is emotionally overwrought after her battle with the Torchbearer, yes, but OTOH the advice she gets about the "Vulcan hello" comes from Sarek, who may occasionally have lapses of judgment but can certainly never be accused of being emotionally overwrought, so it's not as if the idea is something Michael (or the audience) can or should dismiss out-of-hand.
That sparks a series of events that ends with her captain killed in front of her. Which causes he to make another selfish decision in her killing T'Kumva instead of capturing him thus giving the Klingons a martyr for their war. In that moment she breathed life into that war.
After her failed mutiny, though (and Georgiou's at least provisional forgiveness for it), the "series of events" was out of Michael's hands. After all, it was Georgiou's decision for the two of them to undertake the mission to the Klingon ship. (And the fact that
only the two of them went was surpassingly stupid, but not Michael's fault.) Michael setting her phaser to kill rather than stun T'Kuvma was an impulsive act and a misjudgment, but hardly an unforgivable one under the life-or-death circumstances. (And given that T'Kuvma suffered a fatal wound but didn't actually dematerialize, as is the usual with phasers, it's not even exactly clear what setting she was using.) Regardless, the war was already up and running at that point, and any attempt to stop it was a desperate last-ditch effort. It's just that that effort failed.
Michael shouldered the blame afterward because she felt guilty about Georgiou's death. The thing is, that death — never mind the war overall — wasn't really plausibly her fault. Had she chosen to defend herself (in that weird impressionistic Starfleet star-chamber hearing), she arguably could and would have been exonerated.
IOW, the writers muddied things up. If the story was (ever) about Michael directly causing death and war by subverting Starfleet principles, that version fell by the wayside, because it's not what we got on screen.
However after being stripped of rank and being on the Discovery placed in positions where she again has to care for and guide others (tartigrade, Tilly, Ash), having to regain trust (Saru, Stamets, Detmer), seeing others do things the wrong way (Landry, Cornwell, Lorca, Ash) and how far others are willing to go (Lorca) and her disagreeing she learns the error of her ways.
To some extent, she did have that arc over the first half of the season. In particular, mentoring Tilly seemed to restore a sense of perspective for her. However, I disagree about what lesson (if any) she learned from Lorca, since "how far he was willing to go" really wasn't directly analogous to anything she had done... and moreover was mostly presented as being at least arguably justifiable and effective (notwithstanding his eventual MU heel-turn reveal).
Moreover, that same MU episode cast a lot of doubt on what she had really learned about her past mistakes, because Michael immediately cast her lot with the Emperor (despite her being a manifestly untrustworthy mass-murdering cannibalistic sociopath), because she resembled her dead mother-figure, and against Lorca (despite all the good he had done for the UFP, as just mentioned), because she felt personally betrayed by him. This seemed to me a decision driven
entirely by personal emotion, not by any Starfleet principles, much less by any informed perspective or awareness about MU politics and the larger stakes involved.
It ends with her believing more what Starfleet and the Federation teach.
Those values were given lots of lip service in the finale, yes. It would have been a lot more
convincing, though, if those values hadn't been shown, on screen, to (in the beginning) have led the Federation blindly and naively into an avoidable war, and (in the interim) to have been losing that war badly even against a less powerful, strategically disorganized, and politically divided adversary, not to mention (in the end) being something all of Starfleet's and the UFP's higher-ups were themselves ready and willing to toss aside when their backs were against the wall.
So again, the show wound up being thematically incoherent, and Michael's character arc inconsistent at best.
