Why is glycolysis considered central to carbohydrate breakdown?
Glycolysis is central to carbohydrate breakdown in modern organisms, since fermentation is the primary source of energy. Most environments are temporarily anaerobic, and some are permanently anaerobic. Glycolysis can proceed in both.
Once you oxidised glucose, you get pyruvate, which still contains quite a lot of energy. Further oxidation in the citric acid cycle is needed to release that energy.