Show me any evidence that there is ever a lack of oxygen in muscles in high intensity exercise?
It has been argued for more than twenty years that lack of oxygen is not the reason for lactate production, Brook's lactate shuttle hypothesis 1984.
Why do we breathe harder, why do lactate levels increase with high exertion?
Because core temperature is rising during high exertion and we need to recruit more muscle fibers to maintain the pace. As the workrate increases, Pyruvate entry into the mitochondria cannot supply all the fuel fast enough to meet the energy demand, and Lactate production and its oxidation increases.
As for Anerobic respiration, this term cannot be applied to lactate prodution, since lactate cannot be respired anaerobically, it can only be produced anaerobically.
As for reading basic chemisty books, that is not usually a reliable source of information, since the text nearly always contains many flaws.