To be fair, Burnham's supposed to be a bit immature and make questionable choices, which in itself is really annoying to a lot of people, but a product of the plot, and another episode or two and it might might sense.
Also, like @ValueSubtracted@startrek.website, I am a little confused by the "xenophobic military propaganda" bit. Maybe it's that you haven't watched the 2nd episode yet, but without spoiling much, Burnham's somewhat xenophobic actions go pretty horrifically wrong the next episode and basically cause much of the season's plot. Also, the Klingons aren't entirely villains throughout the season and show.
I guess it worked, too; she went from ensign to captain in no more than 16 years… in a gold uniform.
Heck, depending on when she got the Archimedes, she may have beaten Riker to captain.