I am a physician, so I can say this with a high degree of confidence.
That snippet is referring to the circle of Willis*, which is a "backup" circuit that can route around a blockage to the blood flow to the brain on one side.
The thing is the circle of Willis is tiny and near vestigial (there is a substantial fraction of the population where it doesn't even make a full circuit), whereas the carotid is one of the largest blood vessels in the body. The circle of Willis isn't nearly large enough to reroute all that flow. It has to be made bigger over time
through a process called collateralization, and that's a process that happens over months to years, not minutes or seconds.
In short, the circle of Willis will save you from years of high cholesterol that lead to a huge cholesterol plaque completely blocking off one of your carotids. It won't save you from your carotid being severed by a bullet.
*And some other tinier vessels, but mainly the circle of Willis