It was more about being Sarek's daughter.
Except that that doesn't work either. Because nothing about Sarek being Sarek adds anything to the story that random Vulcan #6 couldn't also fill. Here's the thing though, if they did put some thought into it and did want to be fan-servicey there is a much better option: T'Pol.
The entire show is draped in matriarchal themes and imagery. Burnham's entire arc over the first three seasons can be summed up as searching of a motherly figure to coming to terms with her actual mother to ultimately becoming a mother figure herself. It's a nice and steady thematic through-line.
But strictly speaking to the first season, it creates a nice thematic triangle. On one end, you have the pragmatic adoptive mother. On the other end, is the sentimental and passionate surrogate mother. And in the middle is the dark bastardization of both. Burnham takes the advice of each in turn, and it always leads to failure until she realizes she needs to take what she's learned from all three and carver her own path. In the process she defeats two evil patriarchs, installing a new Klingon leader who they call "Mother."
With the Sareks out of the way, the entire focus of season two shifts to Burnham and her relationship with her actual mother. She has to come to terms with the false idolized image she had as a child and realize her mother is also prone to major fuck-ups. And that's okay because we learn from them and grow. It's what separates us from artificial superintelligences.
Then season three shows Burnham realizing she needs to take everything she's learned and take up the mantle of surrogate mother of the lost crew herself - realizing in the process she can't just be the big sister anymore. For one, she has inadvertently and unintentionally stunted Tilly's growth by not being the grown-up.
Because what I think this show is really
meant to be about is exploring and defining what it means for women to be leaders in a franchise dominated by men (And do so to a degree Voyager never came close to matching.) And for that T'Pol has a significant ready-made advantage over any rando. The problem is those is that since t's been so haphazardly slapped together with such little care given to the bigger picture that I'm not even entirely sure this is true. I mean I genuinely don't know if the choice for Orions/Osyraa was a deliberate choice because they're a matriarchal society serving as a thematic foil or because they're green.
