No, EX primals are thankfully not part of the main storyline, and neither are the original 3 Hard Modes. There is some minor storyline associated with them, but they are mostly used as framing devices.

If you do ever go after the EX primals, you will have to clear the first 3 in order (Garuda > Titan > Ifrit) before you can do Moogles and Leviathan (which don't have any order).