If the COVID dynamic resulted primarily from picking up mutations, then the dynamic would be different from country to country and region to region - because mutations occur randomly and because there is no substantial driver towards (no selection for) reduced virulence for a virus with a long pre-symptomatic incubation period during which transmission occurs, as with COVID.
The reason all countries follow a similar dynamic is because they all, at some point, implement public health measures. At that point, or at a point shortly after that, the dynamic changes.
The longest run has been in china, in which public health measures led it to be, finally, eradicated.
If you want to know why I say any of the above, feel free to ask and I will elaborate.