To address your question, Mother Nature is clever. She gives us back up mechanisms, in some cases, lest we suffer a failure of a organ/system that is key for survival—a failure that would otherwise be catastrophic. In the case of the blood supply (vascular system) for the brain, that backup system includes sufficient anatomic redundancy to provide for maintenance of collateral flow to the organ if one supply line is blocked. There are normally 4 arteries, not just 2, that feed the brain: these include 2 internal carotid arteries (ICAs) and 2 vertebral arteries. Your Mayo illustration omits the vertebrals. The key is that there is a central anatomic “manifold” called the Circle of Willis (CofW) at the base of the brain, surrounding the pituitary gland. The CofW is fed by the 4 big arteries, and it is central to the back-up system. There CofW then gives off small penetrating arteries that go on to provide O2 and nutrients to the white and grey matter of the cerebrum and cerebellum. You can look it up. Google can be your friend, but Bing or Duckduckgo should also work.
I think you will find that it will tell you that the carotids can not infrequently get plugged after many years of having high cholesterol levels, just like coronary arteries (etc.) can. As far as I know, the vertebrals are less prone to that plugging (but they can become lacerated—a development that can lead to death—by some vigorous chiropractic manipulations).
As for the carotid plugging by cholesterol deposits, some vascular surgeons can make lots of $$$ cleaning them out. A discussion of the ins and outs of that process is a bit much for me to get into here, but suffice it to say that back in the good old days, the carotids would frequently get severely plugged after years of eating excess lard and ice cream. These days, now that we have statins, etc, to use, I think complete occlusions are less common, but occasionally someone will turn up with one (occlusion = blockage). And FWIW, there are people who have lived for years with asymptomatic complete occlusions of an ICA.
So the folks here who claim that you always need both R and L carotids fully open and working in order to survive are misinformed.
And since you like the Mayo, here is a webpage you may find helpful
https://www.mayoclinic.org/tests-procedures/carotid-endarterectomy/about/pac-20393379
If you want more info, and are in Minnesota, my old office-mate of mine (Hamid) was still working there in Rochester—last I heard—and he could probably do a much better job at explaining all this than I can. Don’t know whether the clinic would charge for his time tho’. They might. I know they charge quite a bit for their CME courses.