Burnham is raised by Vulcans which is used when it suits the plot (then she’s extremely analytical and objective) but then suddenly turns into an emotional mess when they want to portray drama. That just feels … off
No, that part feels very, very on to me. Burnham was a human girl who was witness to family being brutally slaughtered (as far as she knew/could tell) who was then placed in the care of a Vulcan man who liked living as a sociology experiment. This is a person who is traumatized from a relatively young age, and who has no idea how to cope with her feelings. She's never received therapy, only more psychological abuse.
The issue I ended up having with the show is that the show itself never addresses this. It's actually pretty clearly the setup for the entire series, but no one ever acknowledges that Michael needs help, no one ever tries to get her any, and, in the end, she never gets the help she needs. They took what could and should have been a character arc about healing from abuse and just turned it into "SMG's pretty good at crying".
Once it became clear that the show had zero interest in examining its inciting premise, I lost all patience with it.