Zombies dont have zero brain activity, they have limited brain activity, which is why it generally takes a serious head trauma to stop them for sure.Yes, but those require life to perform and the undead brain lacks all of it. No blood flow, no oxygen, no electrical activity.
Unless of course some parasite is responsible for the electrical activity. But atrophy would severely limit any muscle activity outside the recently deceased. Most would be recently deceased, but in a matter of weeks wouldn't be worth a damn.