You're confusing what we know, with what the characters know.Michael Burnham being Spock's sister is farfetched. We knew more about Spock's family than any other character's family in TOS.
No, Spock tells Kirk that his mother was a teacher and his father an ambassador. There's no indication that Kirk knew this. And he seems to forget by the time of Journey to BabelKirk talks about Spock's father being an ambassador and mother being a teacher in This Side of Paradise.

Journey To Babel said:KIRK: What can you expect from a simpering, devil-eared freak whose father was a computer and his mother an encyclopedia?
SPOCK: My mother was a teacher. My father an ambassador.
KIRK: Your father was a computer, like his son. An ambassador from a planet of traitors. A Vulcan never lived who had an ounce of integrity.
No, he didn'tMcCoy knew that Spock's family was important, but not how important in Amok Time.
Gee, another thing Spock failed to mention.Amok Time said:KIRK: Bones, you know who that is? T'Pau. The only person to ever turn down a seat on the Federation Council.
MCCOY: T'Pau. Officiating at Spock's wedding?
KIRK: He never mentioned that his family was this important.
Both of which first come up in the episodes they were introduced and everyone is surprised at the information.We knew that Vulcan marriages are arranged in childhood which makes Sybok's reveal in ST5 logical. Hell, we even knew that Spock had a childhood pet.
Who's to say it wasn't? Especially after they find out Spock's father is Sarek.Spock having a human sister that went to the academy, was in Star Fleet, and led a mutiny would have known as well even with the order not to discuss Discovery.
Secret from who? Us? Everything's a "secret" from the audience until it hits the screen. A secret from the other characters? Not really. It's not like Spock was hiding these things from his crewmates.One secret sibling was enough.