Readers are pushing back on the idea that eliminating sharks might somehow benefit ocean ecosystems by countering the overfishing of smaller species. An ecologist writes:

Sharks (along with other apex predators) play a crucial role in shaping the entire ecosystems that they are perched on top of.  They affect the numbers and behaviors of their prey species, which include many mesopredators (non-apex predators like rays and skates).  They necessarily switch between prey as different types of prey resources become more or less abundant, regulating these species' populations, as well as those of many herbivores, etc.  In doing this, they prevent middle-level specialists from getting too prevalent and wiping out their often more specific prey/foraging material in turn. 

Thus, apex predators like sharks beneficially stabilize their ecosystems from the top down, and promote biodiversity.  An easy example:

Without apex shark species, we've seen skate populations surge on our Atlantic coast to the point that they are threatening shellfish populations with elimination in some cases. Talk to someone who's business is scallops, oysters, or clams about the skate population explosion, and you'll get an earful – these organisms that used to be held in check by sharks are now ridiculously common and threatening some of the base levels of the food chain in their environment, which also means they are badly hurting one of the few remaining and relatively sustainably regulated seafood industries we have left on the East Coast.

He also passes along a guide (pdf) for readers to learn more about the importance of sharks. Another reader:

Yes, removing sharks will allow the fish they eat to grow in number. But this ripples down the tropic levels of an ecosystem: the fish will now eat more, possibly too much, and the result can be the complete collapse of the food chain via a trophic cascade.  California Sea Otters control Urchin populations, and if left unchecked they can ravage kelp forests. Wolves are now understood to be so important in regulating herbivores like deer that they have been reintroduced into places like Yellowstone National Park (with a subsequent increase in community diversity).

Another elaborates on what happened in Yellowstone:

When wolves were exterminated from Yellowstone in the early 1900s, an explosion of deer and elk followed. These large elapids went on to denude the forest of younger trees and saplings, which in turn led to a lack of habitat for songbirds. It led to a boom-and-bust cycle of starvation for the overpopulation of elapids. It led to a lack of raw materials (young trees) for beavers, which adversely affected the fish that were dependent on beaver dams to create ponds. The lack of fish went on to hurt other predators that were dependent on fish as a food source. When wolves were reintroduced, this cycle reversed. 

So this is not an obscure theory; it is taught in every basic Bio course in every college in America. The sharks are known as a "keystone species," which helps regulate the health and function of the entire ecosystem.