No, that argument does not hold. The rate of mutation is proportional to the number of infected people. Every person infected over the course of the virus' sp

No, that argument does not hold. The rate of mutation is proportional to the number of infected people. Every person infected over the course of the virus' spread is a single roll of the dice to see if a new variant arises. If X% of the population gets infected over the course of the virus, then you get the same number of mutations/variants regardless of the timeframe. Letting a disease spread rapidly just means that those variants all arise over a shorter period of time. The way to decrease the number of variants that emerge over the course of a disease is by decreasing the total number of people infected over the full course of the disease. That requires widespread vaccination.