That itself is the question: since when does anyone cook in a sink? No one disputes that there is an obligation to kasher a utensil used for eating or cooking, but a kitchen sink is not included in that. Is there also an obligation to kasher a chair that became non-kosher, such that as long as the chair has not been kashered it is forbidden to eat while sitting on it, even if it has not been used within twenty-four hours?

It has nothing to do with cooking. If the Sages decreed rabbinically regarding the emission of absorbed taste even when it has not been used within twenty-four hours, that applies to a sink too. Unless you assume that in a sink only spoiled taste is absorbed all the time—not because of the passage of time, but because we are dealing with leftovers. The custom is based on the assumption that this is not the case.