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 think episode 8 is worth a shot. The basic premise is really stupid, but they execute it so darn well with the writing of social interactions. Also, I'm really surprised they pulled off Vulcan Patton Oswald, which initially scared me when the season trailer came out.