Oh sh*t, you're right! I forgot about Presence of Mind!
So WHM, BLM, NIN, MNK, SAM, and maybe SMN and MCH(because of Dreadwyrm/Firebird Trance and Heat Blast due to how they work) have access to haste but none of the tanks and Physical Ranged do not have access to haste, so at least 1 tank need to have access to some form of haste...
EDIT: WAR Fell Cleaves and Decimate also qualify as haste effects because of how it affects Infuritate recast timer, but that's on a technicality like Dreadwyrm/Firebird Trance removing cast times and Heat Blast affecting Guass Round/Ricochet recast timer...