First, it's not just the shuttles in one episode. TOS contradicted itself all the time. James R. Kirk. Space Command. Star Service. 'The only death penalty left on the books' (which was stated about multiple different regulations). The Mind Meld, which was never used in situations where it obviously would make sense, until it was with the excuse that it's very 'private', but then they just use it willy-nilly for the rest of the show. Spock smiling and making jokes (like, a LOT). Etc, etc, etc.
Second, Burnham may be 'famous' as Starfleet's first mutineer, but her connection to Spock obviously isn't widely known so it's ridiculous to call that a problem. And Spock having an adopted sister in his youth (who was not always around, since she had her own life before the attack that killed her parents) is no bigger than every other revelation that's ever been made about Spock's past going straight back to Journey to Babel. The simple truth is, Spock doesn't talk about his past, even with his supposed friends. They didn't know his mother was human. They didn't know his parents were going to be passengers on the ship until they arrived. They didn't know he was engaged/married until Amok Time. They didn't know he had a half-brother who rejected logic. The most enduring truth about Spock is that nobody knows shit about Spock.
