No. They can make use of glucose or ketone bodies. I recommend a biology textbook before commenting on fairly basic and long-established facts.
However, even if they only metabolised glucose, the brain's requirement is for a constant but low level supply. It doesn't need to be provided by a dietary source; the body will synthesise it on demand from its energy stores. The cancer will need to compete for that small quantity with the rest of the body, which will limit its growth.
When you eat sugar, the body can't immediately store it since it takes time to do so, and blood sugar levels spike for a period before reducing back to baseline levels. During this time sugar is in great excess in the bloodstream and tissues, and the cancer is able to utilise as much of it as it can, which can result in its survival and growth.
The survival and growth of a cancer is largely based upon various kinetics, proliferation rates vs death rates, and the evolution of a cancer through selective pressure over time. One of the key switches is to anaerobic metabolism. It lets them switch to lactate production to survive a lack of oxygen. But it comes with a limitation: they are restricted to glucose and other simple sugars to survive. By strictly limiting their availability, it can have a significant effect upon tumour viability. Until it undergoes another change under selective pressure, of course, but it's a simple and potentially very effective course of action, without any great risk of harm.