Clearly, the makers wanted to do something bigger and bolder, but it didn't end up as being epic as it should have been. That's from recollection.
But, nope. It looked like she wanted to shoot the Klingon officer instead of capturing him, per the orders she was supposed to obey. And, of course, the mutiny stuff with Captain Georgiou - well intentioned, but faulty and not because it's ostensibly called "the Vulcan hello", and I'll return to that in a moment. Never mind the lens flare as the exposition was given since the director preferred to distract the audience with pointless JJesque eye candy instead of her insistence of what this show's interpretation of Vulcans are like. Never mind a lot can change in 240 years despite basic emotional processes, otherwise humans would never have evolved in the first place. But I digressed; if the script had not been at rough draft stage when they filmed it, and camerawork directed a bit better to avoid those distracting lens flares (they're selling the plot as conveyed by the acting and not the cinematic special effects cutting in, surely?), they might have been able to prevent the wide range of beliefs the audience members cultivated more readily. I've no major inkling to see it again.
And yes, the Klingons were already wanting to start a war. Is that a big game changer for Burnham? Not necessarily. If anything, her impulse actions may have only accelerated events, which nobody wanted as Georgiou was trying to prevent a war and a war is something like a really big significant thing, preferably to be avoided. Neither character was wrong as such, Burnham's "Hi there" perspective was told plausibly as it actually fits into Klingon belief rather reasonably well for this show, and yet Georgiou still seemed to understand a bigger picture. Burnham was talking of just individual skirmishes, centuries earlier, before there was a Federation. That might be why Georgiou was in command, "We don't shoot on a hunch". And imagine if Spock convinced Kirk of doing Vuncan howdies all the time (but that's another show, one where Kirk was in command and Spock still realized it's not always about logic, as shown in numerous episodes, the more heavyhanded example being "The Galileo Seven", but I digress. ) Burnham makes an
amazing change within the same season since she went from being the most incredible officer from most dire of contrived circumstances to being the most incredible officer ever, also under the most contrived circumstances... All in a dozen or so episodes, that beats everyone else's record by far!
Thankfully it's in its own canon, whereas Kelvin is its own timeline and Kirk's in his own. There's no way to keep strict continuity and the debate over our being told "it's prime" either means it's supposed to fit in with Kirk's (which is proven to be impossible) or DSC is the new official timeline and the others are their own, separate timelines - like the theory of infinite universes where each action is taken and the all results form their own realities (see the 1995 TV show "Sliders" for more.) So remove stern continuity and accept DSC on its own merits and failings and any perceived problem about canon is solved. Even the one where they're using fanservice as plot points (mirror universe so quickly, etc, etc.) What remains are fair issues about plotting, acting, et cetera.
There's a
far more cynical article that was made at the time the show premiered:
https://www.irishtimes.com/culture/...am-i-shout-at-the-screen-in-klingon-1.3234222