All of these connections make sense to Burnham in-universe. The only reason we care about Spock and Sarek is that we already know them. Otherwise, they are largely insignificant on a universe stage.Notice something about these?
They make in-universe sense, because there is a plausible story reason in each case.
And yes, these were already stretching it even back in the 60s.
But even they don't manage to come close to Burnham being the most important person in live to 1) Spock, 2) Sarek &Amanda, 3) Both the Universe Emperor and her biggest rival and now the chief of the most clandestine spy agency in the Federation also being personally involved in her tragic backstory.
Look, I'm one of the people that like Burnham. But that is absolutely because of the actress and her day-to-day acting, NOT her beyond-ridiculous fan-fiction backstory.
This is a general concept of fiction. It's easy to swing the other way and zero connections but then those connections come across as forced in order for the action to occur. What's actually kind of interesting about DSC is that it has taken some of the familial aspects that many liked in other shows and woven them into an actual family, with all the drama that entails.
It truly is no different than the mirror universe stories where Kirk can take on the role as his counterpart to try and undermine the Empire, or Mirror Bariel coming to ask Kira for help, and on and on.