...Can someone please explain to me how that seems like good game balance? It seems to me like zombies get pretty much every advantage at later levels.
It's true, zombies basically become low-tech Terminators once they have all the skills. The reason for this is twofold:
One, it helps compensate for the lower number of zombies. For various reasons, not the least of which being that people just naturally prefer playing survivors, usually survivors outnumber zombies. In most zombie movies or whatever the main reason zombies are so menacing is that they vastly outnumber the survivors. When that isn't the case, they need to be made more menacing somehow. Which relates to point two...
Two, in a game about a zombie apocalypse you want the combat odds to be tipped in favor of zombies. Zombies should have an aura of menace about them, and you shit your pants every time you get into a scuffle with one. "Balance" in this case isn't about both sides having an equal chance of victory in battle, it's about the overall fun level of the game. While zombies are just straightforward killing machines (well the one exception is that they can gain the ability to speak sort-of, and so zombies sometimes moan amusing quips when they break into a building and see survivors,) survivors can do all sorts of things besides combat to help them survive, such as barricading, healing, and reviving. Really, combat is about the least useful things survivors can do, in large part because zombies have such an overwhelming advantage in combat. Unless you have a specific reason for attacking a zombie (you're clearing out a building, or you just need the XP) then you should always run.
To put it another way, the game is not setup to reward players who try to be a badass (those players often dress their character in a black trenchcoat, and thus experienced players derisively refer to them as "trenchers.") The game rewards players who actually do sensible things to try to survive (ie. run away from the zombies.)