Kurtzman has said they are allowed to use material from all the movies, including the Kelvin ones. And indeed, we have already seen stuff from the Kelvin movies in Disco like starship bridges having windshields and the Trek XI auditorium. So, they can obviously use anything from any of the movies, despite what Harberts and Berg may think.They never specified what movies they could or couldn't take things from though, and considering the circumstances, they probably meant the Kelvin movies as something they couldn't take from.
Michael was what, seven or eight when she began living with Sarek? With her being "a few years older" than Spock (according to Mack's novel) and Spock being about five or so when Sybok was exiled (based on the deleted scene), there really isn't much time she and Sybok could have spent together.If anything Sybok would be the one at thanksgiving dinner who could tell a twenty minute fascinating story about his last teeth cleaning.
If Sybok's behavior was the same through-out his life as we saw in TFF, then he would be the "most Human' of Burnham's adopted family. For this reason Burnham might have had a easier time interacting with him, certainly when she was still a young child.