Since zombies do not feel anything, it would be impossible for them to starve to death since, essentially they are simply a virus. Only living to spread and infect all of those who are not infected. Also, zombies do not only devour the brains of their victim, but the entire entity, possibly leaving behind bones and such.
Just kidding, I shouldn't say that. I would say that if a zombie can't find brains then there isn't even flesh to eat, and since zombies kind of thrive on that, its body would break down and it would become a pile of rotten flesh. Hope that helps. Thank you.
Zombies are already technically dead. I think brains are just their source of protein which makes them just be more lively and awake. Without brains they are groggy and tired. It is essentially just like how humans are without food, except without food humans would die. Zombies are already the dead among the living.