Just because you're a carrier doesn't mean that it's strong enough to overtake you. We all carry small diseases in us I'm sure, but our bodies can fight them off and don't require the need to make us feel ill, or are overtaken by them. The bite from the Walkers come from the disease being too over powering probably, which is why healthy people take longer to turn that either extremely injured people, or dead people, who have no immunities at all. Could also explain how an airborne disease only overtakes a few when it first takes shape and infects.
In short, the virus infects everyone, but only a few don't have the immune system to fight it off. When bit by the Walkers, the disease is strengthened, and body can't fight it back. It easily explains why virus started slow and grows and grows.