Yeah, I don't get the sermon thing. I mean her personal logs just seem to be her...personal musings on the situation. Thematic exposition rather than straight boring exposition.
Although it seems people want more of the latter: "We're currently in orbit around/ en route to [insert planet] to [insert mission]. The inhabitants here have problems with [possible disaster or conflict]. The situation is [insert exciting adjective], but I have every confidence in my crew to resolve this with minimal casualties." Rinse repeat.
I think the only time Michael has sounded lecture-y is when she's addressing Starfleet personnel in a public speech (like the award ceremony) or when she has to publicly rebuke Starfleet (re: Cornwell's approval of genocide). Because that's the point of her story arc, realizing where she went wrong at the Battle at the Binary Stars and wanting to return to the Federation ideals she believed in.
I find her far less self-righteous and sanctimonious, for example, than Picard or Janeway or Sisko were in their moralizing speeches. And she's a lot less hypocritical than the latter two, imo. Picard had this "we're more evolved than you aliens" air to him which, upon re-watches, now irritates the heck out of me.
I think Burnham's monologues are essentially what she talks about in the opening episode of this season: an exploration of the human condition. The journey of self-discovery is a big focus of this show.