I don't buy into the notion that every dead body comes back as a zombie, was it stated in an episode? There were too many bodies outside the hospital in the pilot and the CDC that had not had their brains mangled and were not zombies.
But lets run with that scenario anyways, just for fun.
Lets say that major cities and towns get overrun within the first twenty four hours. So what? Every major military post is fine. Each one has thousands of combat troops, 30-90 days of food, fuel and ammo, self sustaining water production. Miles and miles of stockpiled razor wire and other barriers.
Plus, almost every rural house in the USA has at least one firearm with some ammo. By conservative estimates there are over 300 million guns in the USA, call it one per person. Billions upon billions of rounds of ammunition.
But the survivors we get in the show are a diverse group of idiot rednecks, academics, hippys, and a pair of cops? (Oh, and don't get me started on the Hispanic janitors guarding the retirement home) With what, 3-4 guns at the start, and a seemingly weak survival instinct at best? Those are the people who survive the zombie apocalypse? Hell no.
The zombies couldn't get into a department store for seemingly hours, and you want me to believe that they overran the military and all those kooky militia groups out there?
I have a really tough time buying into that premise.