Indian Cobras
"Rewards are given to those who meet a giant snail hunting quota."
During the British Raj a bounty was placed on cobras to reduce their population. The cobra population exploded because people were intentionally breeding cobras to collect the bounty! So too your snails.
They probably COULD be hunted to extinction... except for the breeders. You see, Snail-bounties can pay enough to permit it to be a full-time, relatively undangerous job. Provided you can hunt enough snails. How to you guarantee enough snails? Why simple, you breed them yourself! Of course this is Very Much Illegal but when you make "maybe my family will have enough food for the winter maybe we won't" levels of money there's a very strong incentive to skirt the law and get yourself a nice payout. Of course the snails are a bit dangerous so you make sure your creche is in some OTHER town's woods/marsh/cave system whatever. Then when they're hatched/the right size you open the pen, slaughter them all, and collect your bounty! Not your problem if your pen is poorly made and some break out, they'll just eat the OTHER guy's crops!
So you end up with a constant low-level "there might be a plague of giant snails/the odd snail eating my field."
Additionally, snail-killing could be a mark of control. "DUKE SNAILSMITER KEEPS THE SMALLFOLK SAFE!" The herald cries, while Duke Snailsmiter ensures there is a constant flow of snails across his realm so his valiant knights can smite them and earn the continued fealty (and taxes) of the peasantry.