A couple of things that are apparently needed for a successful liver flush is fats and acid.
Olive oil has the fats and coke has an acid as does grapefruit juice.
Grapefruit juice as used with the Clark flush has citrus acid.
The Idea of using coca-cola classic is that it works, and the palatability of it makes it a little easier to get down larger amounts of olive oil.
You really have a choice, coke or grapefruit juice.
Of course anyone who participated in the "Fear Factor" eating events would probably laugh to think anybody would need Coke to get down a few ounces of olive oil.