I'm not sure I see in the image what you can see in person, but generally one wants all surfaces above the burners that face the burners to be metal or ceramic or some material that is inert and cleanable. I would want an insert (liner) to have flanges at the bottom that cover the cabinet when the liner is (conceptually) raised into the cabinet. To successfully use exposed wood, the wood would have to be well outside the expansion zone of the rising hot effluent; i.e., the liner would have to have a large aperture.

Perhaps your cabinet fabricator thought your kitchen was only decorative, like many kitchens illustrated in magazines.

kas