I think that's the issue right there. They created a strong backstory for the character, along with an intriguing background, but then they've done very little to build on it. What little we've seen of her relationship with Sarek seems like a variation on the theme of the Spock/Sarek relationship which was well-explored in TOS, the TOS movies and TNG. We have yet to see her deal much with her own feelings about how her convictions about how to resolve the initial conflict with the Klingons failed badly and, arguably, did help ignite the war.
I also think that the life sentence was a mistake. For me, I see the Federation/Starfleet penal system as more enlightened. The two times we saw it in TOS, extended prison sentences were explicitly stated to be the exception, not the norm. In my view, it would have made much more sense to have her receive a shorter sentence - say one to three years. That sentence could have then been commuted or suspended when Lorca requested her presence on Discovery. Although the reason why she was so necessary to his mission still eludes me. Burnham certainly seems like a highly competent officer, but Starfleet should be full of those. Even during wartime, Lorca shouldn't have needed to pull her out of prison in order to staff his special mission.
I wonder if we may see a time jump between season one and season two. If the writers can't figure out how to show Burnham coming to terms with her past and going through some redemption, perhaps they can just move past it. It could be dealt with from a distance as the theme seemed appropriate in future episodes, rather than seeming like a missing part of the story, which is how it seems to me at the moment.
