Spock's actual lines in that scene are:
SPOCK: The destruction of Discovery was tragic but does in and of
itself resolve the issue. Even more radical steps must be taken to
ensure that type of scenario never repeats itself.
Any residual trace or knowledge of Discovery's data, or the time suit,
offers a foothold for those who might not see how critical, how deeply
critical, that directive is.
What Spock is suggesting here is that any information about the Discovery or anything related to it could give someone (or something) in the future hints that might ultimately lead them to discover the existence of the time suit and it's technology. If the time suit was re-discovered it would threaten the entire timeline.
Therefore, Spock proposes that all possible leads to the time suit be purged/destroyed - that includes all information relating to Discovery, including the Spore drive it contained, as well as the time suit itself. The Federation are not "evil" so they don't can't just disappear all the surviving crew, but they can make them swear to secrecy for the rest of their lives (there are real-world examples of this kind of thing occurring).
Or, as Spock himself puts it:
Therefore, to ensure the Federation never finds itself facing the same
danger, all officers remaining with knowledge of these events must be
ordered never to speak of Discovery, its spore drive, or her crew
again. Under penalty of treason.
Note he doesn't mention officers of the Discovery specifically, it's "all officers" [of Starfleet].
Out of universe this does of course also tie up the huge plot hole, which is that no one in any of the shows set chronologically after Discovery ever mentions spore drives or the Discovery.