It doesn't appear implausible that Starfleet would have a number of classified projects that involve oaths of silence from its personnel. Any big business worth the name today has those. Doesn't mean the secret to be kept would be massively important, and doesn't mean that extraordinary measures would be required to stop leaks. Spock recommended "treason" as punishment/sentence for babbling, whatever this odd choice of words means. Businesses may use different legal jargon. In the end, the so-called secret may still leak out, usually to little effect.
So Starfleet doesn't want folks to know about the research conducted by the Discovery in wartime? That's no doubt standard fare, and happened to a dozen of other things during that war, too. A secret easily kept, as we saw in those S1 episodes where this Lone Ranger ship was a mystery to those she rescued, or left no witnesses (other than Klingons, who may have mistaken the disappearing act for a Starfleet cloak) when she failed.
So Starfleet wants to cover up the emergence of a vicious AI, essentially a screwup by the organization? Again, standard fare. And again, very few witnesses, as the AI killed most of those directly involved.
So Starfleet wants to label "dead" people who are off to other things? Kirk did that often enough, too. And given what our heroes do for a living, them going missing and presumed dead will raise no questions.
Few of these would appear to have longterm consequences for Starfleet or its later sets of heroes. What is relevant is the hush-hush around the existence of the mycelial network and its potential. But this is an exceptionally easily classified piece of knowledge, akin to Genesis technology: tested in great secrecy, unlikely to be accessed through independent research, not a big problem for Starfleet if it goes unused... Genesis at least might have involved widespread research correspondence and a paper trail, but Straal and Stamets would simply get shot for writing anything down.
And Sarek would have much dirtier and more relevant secrets in his head, the contents of which he probably didn't fully share with anybody, least of all Picard. The "my mind to your mind" thing has always had its limitations, the hero melder failing to learn all the facts despite lots of trying (extreme example: TUC).
Would knowledge of Stamets' research have helped Janeway? Certainly. But why would Starfleet care about the Janeway scenario? It's an extremely unlikely one, and total loss of the Voyager would be rather trivial to the organization, a spore rescue operation not worth the hassle. I mean, the organization certainly knows about time travel, but does not engage in constant resets anyway, these again and demonstrably tending to be a worse disease than the one they try to cure.
If anything, the final scenes of "Sorrow II" comically exaggerate the degree to which the secrecy was to be implemented. Just trust Sarek's judgement as a person who knows when to share. I'd tend to trust Picard there as well.
Timo Saloniemi