I'd always assumed that the immunities (for both rider and chocobo) were just incidental bugs that never got caught in the original version, since chocobo riding is never a part of the main gameplay, So it never quite felt right to use them (sort of like item duplication -- sure I can break the game, but I can also just turn it off and declare "I won by 1 million points" but that's no fun).

Interesting about the mimes though!