Her actions did start the war.
Particularly because she disobeyed Georgiou.
Georgiou ordered Burnham to do only a flyby.
She lied to Gereogui with a straight face and disobeyed her order to satisfy her personal curiosity.
That led to her killing the Torchbearer and that drove T'Kuvma over the edge.
Then she failed at stopping the war because she did not capture T'Kuvma.
She did not follow the plan, disobeying orders again.
BTW, she failed at capturing T'Kuvma not because of her inability to do so, but because she chose to fail by killing him.
Lying to her captain, disobeying orders (twice), attacking her captain, and trying to take over command. All of that was in the first episode of the first season.
Yep. Michael had a lot to learn. Season One was all about her redemption from the terrible choices she made. Definitely not a Mary Sue.
And people don't understand why Burnham is not liked.
I mean, it seems weird how one faction hates her for being perfect and the other hates her for being evil and neither side seems capable of accepting the idea that she started out deeply flawed and has become a better person and better officer over time.