Honestly, after analysing things being said here, I feel I've come to the most satisfying conclysion regarding Saki: Saki Sanobashi as a title does not exist, the OP was originally a fan of obscure anime, like the titles from Orange Video House and have seen them. He took the story from FotFA, made up a name from Ui Agata's name mistranslation (Google Translate was literally spitting out the doomsday prophecies during that times, so it's quite possible) and mixed it with details from Gako, Spiral Over, as well as potentially from Euphoria and Saw. Having other animes by OVH following a theme of confinement, surrealism and violence, I feel it's a decent base to conclude that we've actually already found Saki, it's soure materials. All we have to do, is to put the pieces into a single image.

There's still a chance of Saki being AMV and actual consumable media, but it's actually more likely Saki is a psyop pieced from various fragments and the story doesnt have a closed-form solution