"All we need is a giant tardigrade and some mushrooms for every Starfleet vessel!"
Discovery did seed a lifeless planet with those spores... so, they have that taken cared of.
As for finding a Tardigrade... that's not necessary.
Its possible the UFP could make an exception in providing well trained pilots with this genetic modification for the sole purpose of using the Spore Drive... after all, they DID allow Bashir to continue to serve in Starfleet.
But, they wouldn't necessarily need to.
If you recall, the problem of successful use of the Spore Drive was also mentioned to be rooted in lack of sufficiently advanced computer technology (aka, AI which would be able to do that).
Starfleet had artificial brains by 24th century... so its EXTREMELY UNLIKELY that they wouldn't be able to make the drive work.
For that matter, 32nd century Starfleet would similarly be able to use the Spore Drive via technology alone and no need for biological entities of any kind because the tecnology would have evolved to such a degree where technically speaking Spore Drive wouldn't really be needed because Transwarp beaming would have been used for both people and ships (which would in itself render distances useless... unless TW beaming does have a distance limit - but 32nd century power source would provide ridiculously massive ranges most likely... across half a universe probably).
How was it sloppy? The news didn't get out.
I guess it was sloppily handled when it was covered up initially.
Plus, the Klingons were at large closing up on Discovery and its secret in Season 1. Its unlikely to think they would have abandoned that research (especially if Klingon Intelligence thought that USS Discovery could show up anywhere)... although they also decided not to mess with the Time crystals and time travel in general, so I guess abandoning further research into what Discovery had (beyond the Klingons who learned about the Spore drive and were destroyed with the Ship of the dead).
Still, Burnham's ex and the female Klingon Chancellor might have thought the matter closed after realizing what was at stake (annihilation of all sentient life in the galaxy) so they could have decided to eradicate data pertaining to Disco themselves.