Cornwell wasn't in charge of S31. That was the Vulc-cicle from the station. She was in charge of the "red signal" mission. She had access to S31 resources to that end.
We might have thought so before the season finale. But there the death of Cornwell specifically is quoted as the reason there's a job open at the very top of S31.
Seems like a very flat
and narrow hierarchy, too: other losses specifically quoted were Captains Leland and Georgiou. So it seems there were no S31 Commodores, say. Or even Commanders of greater seniority at that rank than the wet-behind-the-surgically-implanted-ears Commander-who-never-was-Lieutenant-or-Ensign-or-even-raw-Cadet Tyler.
But flat and narrow is what you get if S31 kept most of its assets aboard those 30-something ships Control then sterilized, or sent its very top brass and their staff to that former prison satellite Control also sterilized...
As such, the death of the S31 leadership is not something she gets to cover up, especially when it seems like the whole fleet was involved with checking for onboard Control infestations.
Only the S31 fleet checked for Control infestations; the rest of Starfleet knows Control exists, and will notice how Control suddenly stops responding to their user queries, but that's about it. And at the specific moment of her death, Cornwell would appear to have been the last S31 officer of any note standing, apart from Tyler and Georgiou, and barring the miracle recovery of Leland. She wouldn't be covering up for the loss of S31 leadership, she would be getting charged with S31, in all senses of the word.
Accepting that Cornwell always was a S31 boss (even if not as high up as the quartet killed at the prison sat in "Project Daedalus") explains a great many things about her odd comings and goings. The Spore Drive seems to have been a S31 project, as per the black badges in the early episodes, comparable in many ways to the Red Angel project. Cornwell would be the one keeping close watch over the project, and over its point man Lorca who (soon after getting his bearings in this alien universe) no doubt gained access to the ship and its tech through his powerfully connected lover in the first place.
Cornwell's odd appearance aboard the S31 ship in "Saints of Imperfection", after seemingly hiding in the shadows throughout the spore realm adventure, is also nicely explained now. As is her general annoying habit of oddly disappearing.
The only real question here is, what sort of "cover" jobs did these explicit S31 bosses like Patar and Cornwell have? Did they bother to pretend they held some other job or not? Cornwell really does dedicate herself to shady stuff during the first season - but then again, it's wartime and "clean" bosses might be doing shady things, too. OTOH, Pike doesn't bat an eyelid when Cornwell marries him to Leland in a sermon preaching the virtues of S31 in "Saints of Imperfection" - but then again, Pike doesn't bat any eyelids at mushroom drives or resurrections or time crystals, either.
Timo Saloniemi