When I here this sentence I know that it probably means drinking water or some normal liquid but to my family "drink" is not important to have at all. "The Drink" or "drink" is alcohol, more specifically the abuse of it. So food IS more important. Now from a survival stand point water is your first concern then food.
being non-native EN, but with good and long experience with EN, I "feel" that something is wrong with just "drink" at the end.
"Eating is more important than drinking" sounds OK for me, even if sometimes that is not always true either, but it could have lead to less discussion.
Hugo, you are right that "eating" pairs well with "drinking". However, an equally valid pairing is "food" with "drink" (not "drinks").
In the above pairing, "drink" is being used as a singular collective noun. That usage is a bit elegant and "higher register" -- perhaps even a bit "19th century".
It would be unusual to come across it nowadays outside of the phrase "food and drink". Still, educated native English speakers would be familiar with it.