It really boils down to how the virus/pathogen would work, and the nature of the infected wound.
Using your anesthesia example, in that case, it was injected into a major vessel, and thus made its way to the heart in mere moments, which then sent it straight into the brain. With your extremities, the blood has to flow from the capillaries to the surrounding veins, meaning that you have about as long as it takes for that particular area to be served by the flow of pumped blood. Blood takes about 60 seconds to go from heart to back again, which is probably how they figured the 30 seconds.
Although it was probably a bit of creative license for that figure; it'd come down to a split-second decision, no matter how you cut the timeframe.
I wouldn't call it medically sound, since bites are penetrating wounds, and penetrating wounds tend to reach the larger vessels (not just the capillaries and smaller vessels, as with a finger). You do have to keep in mind, also, that blood splatter/gore could just as easily infect, and that such infection could go undetected.
Maybe if the pathogen were actually a bacteria or other organism, then it could be possible to amputate and save the body, as is the unfortunate case of flesh-eating bacteria, or if it were a very specific sort of neural virus, then maybe that could be an option.
Rabies, for example, can take up to several months to reach the brain; if you were infected in your foot, the rabies would have to travel the entire length of the nerve fibers up to your brain. The only way for the virus to reach your brain would be to build up enough viral particles along the fibers to advance, like trying to reach the ceiling of a room by pouring sand onto the floor. In the meantime, you could be injected with the vaccine and be able to train your immune system to kill the virus, before it ever got there. Likewise, if the virus is of this sort, you could theoretically cut off the limb and prevent that from happening.
If it were a virus, one that holds no punches and could begin its damage the moment it reached the bloodstream (i.e Ebola), then I'm sorry to say that I do not believe that it would be useful to cut off the limb. If there is no cure and it has a very high rate of success, then there is pretty much nothing to be done to save them. You could wait, until they could prove themselves either able to fight off the infection or doomed to their fate, but that is up to your discretion in the field.
Basically...if it's rabies or a native variant of rabies, and the wound is not near or on the trunk of the body, amputation may be an option. If it's blood-borne, amputation is a Hail Mary at best, and an albatross of suffering at worst. That's how I see it.