Responses
From the beginning his desire is to get to know monster. How they live including how they taste. Looking at how they said no monster would ever get close to him in a whole town radius, I think the curse is no monster would ever get close to him. (That's why Izutsumi sell him monster carcasses, because he can never see them in their habitat anymore). Kensuke probably died after the curse too and the cocatrice simply couldn't run because it's hurt.
So any theories on what the curse is? The most popular theory is that he won't be able to enjoy eating anything anymore, but I feel like the actual curse will be more complex than that. Like the lion specifically says that his desire will never be granted, not anything about the loss of desire. Idk though, the latest few chapters have shown him in a more subdued manner regarding eating, so that theory makes the most sense. It's also not the angst fest that is the Falin not reviving theory, so I'm personally hoping that it's true.