Steam only checks cloud saves when a game is run and it only cares if the save files on the system are older than the save files in the cloud.
But in any case, I decided to try something. I disabled cloud saves, made a brand new character (SC/Normal), played until I killed the reanimator and then visited the illusion vendor. Multiple pages of illusions. Closed the game, ran GDStash, clicked Add All Illusions, closed GDStash, loaded game and once again, only a single page of Illusions.
There are also no backups dated at the time I did this, so if GDstash is supposed to what’s making the backups…it’s not, or at least it didn’t in this case.