I feel like that's a misinterpretation of the character. I agree that she's not written well but Burnham is not only supposed to show emotion, she's the one who teaches (and reminds) Vulcans that it's okay to have them. She also serves as a counterpoint or foil to the emotional extremes of Mirror Georgiou (hostility, brutality), Tilly (excitement, naivete), and Saru (duty, strictness) -- if they're at one end of the extreme, Burnham is either at a more reasonable end or at the other end of the extreme.
My main problem is the writing. Burnham gets to emotional highs and lows but oftentimes the writing doesn't work to earn those emotions, so it seems like she's crying for no reason. On paper the progression makes sense but the pacing on screen is so off that those emotions seem more spontaneous rather than natural. For example, her sadness at a colleague's death would make way more sense if they actually devoted time to that connection with the colleague, rather than at the episode of the death itself.