They're so dumb that they really shouldn't be a problem in a few decades or so. The natural world is so full of all kind of lures and hazards that the zombie population would realistically be in a sharp steady decline.
Walkers have no ability to discern or avoid danger and are relentlessly drawn to the smell of living things, as well as sound and motion regardless of the source.
Consider an urban center like Niagra. What famous natural feature there do you think the local dead would wander towards and get destroyed in? Might make it one of the safest places in the world.
Consider white-water rapids. They're noisy things that switchback through the countryside, and anything caught in them is torn to shreds.
Consider the sea shore, particularly during rough weather. Ocean waves are loud and powerful and don't care about walker bites.
Consider winds howling around a cliff, drawing walkers forwards until they tumble from the top like lemmings tossed to their doom by a Disney filmmaker.
And while they only "die" when the brain is destroyed, there should be a rapidly rising percentage that can only move by crawling around. They might not care if they shatter their femur in a fall, but they still won't be able to stand on it. If the story were ever to skip ahead a decade it should probably be retitled "the dragging dead."
Maybe there are six billion walkers and less than a million humans left in the world. But the dead's mindless aggression towards noise, movement and smell should be making huge in-roads towards bringing their numbers down.
Passed out for two days in a car wreck, leg lacerated and surrounded by zombies.
First order of business? Bourbon.