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.