Certain cells, such as our red blood cells, lack the mitochondria to be able to metabolise fatty acids or ketone bodies, so they require glucose (which can be metabolised anywhere in most cells). I have also heard that certain parts of neurons are too narrow to contain mitochondria, so they, too, require glucose (though I have also heard this notion challenged). In any case, if we eat no carbohydrate at all, our body maintains a low level of glucose in the bloodstream, and our muscles and liver maintain a certain amount of glucose stored in the form of glycogen. But the circulating amount of glucose is quite small, if the body is left to manage itself.
My understanding is that glucose metabolism is far older, evolutionarily speaking, than fatty acid metabolism. The presence of mitochondria in our cells is a form of symbiosis between the cells of our body and organisms that used to be separate. (This is known, because the mitochondria have their own DNA, separate from the DNA in the nucleus of our cells, and mitochondria with different DNA will fight and kill each other, and the cell containing them.)