Well, it's not a prequel in the third season. It takes place in the 32nd Century.
At the end of the second season, in the season finale "Such Sweet Sorrow", Discovery jumps 930 years into The Future with intent that they never return. At the end of the episode, those who remained behind in the 23rd Century, including Spock, recommended that Discovery and the Spore Drive be classified under the penalty of Treason and that they never mention Burnham again.
Classifying Discovery and the Spore Drive are things I can understand and can rationalize in-story. Their decision to make it so Burnham isn't allowed to be mentioned again, that just reeks of trying to satisfy fans who won't like anything
Discovery does anyway. It's the one part of the episode I didn't like. I'd have rather had the Spore Drive still be classified but have the Official Record just say Discovery and the crew were MIA and leave it at that.
When they figured out how to explain why Burnham, the Spore Drive, and Discovery were never mentioned in TOS or any of the series that take place afterwards, what they came up with was, to quote a line from
Mad Men, "the only [solution] that's equally offensive to all."