Spock didn't speak of Sarek until he appeared on the USS Enterprise one day, nor did be speak of Sybok until he was taking over the ship....Spock just doesn't talk about his family. Michael, being family, would be someone Spock doesn't talk about unless she actually shows up in front of him. He also doesn't talk about his childhood much, likely because it is embarrassing or mildly traumatic, though he did mention his pet, but mostly in context with another animal that was nearby as a reference point.
And while Captain Kirk might know of Michael Burnham, he might not know her ties to Spock, unless he knows of Burnham's connection as a ward to Sarek of Vulcan, and piece two and two together after finding out that Sarek of Vulcan is Spock's father. Because while that information might be in Spock's service record....usually no one cares who your parents are in Starfleet unless you happen to have a famous sounding last name. Which is something we can't say that Spock has, since he goes by just "Spock".
And while Captain Kirk might know of Michael Burnham, he might not know her ties to Spock, unless he knows of Burnham's connection as a ward to Sarek of Vulcan, and piece two and two together after finding out that Sarek of Vulcan is Spock's father. Because while that information might be in Spock's service record....usually no one cares who your parents are in Starfleet unless you happen to have a famous sounding last name. Which is something we can't say that Spock has, since he goes by just "Spock".