They seem to be ignoring, or downplaying to the point of not mentioning it, the evolutionary pressure we are putting on the virus/human organism.
All they seem to be concerned about, with regards to mutations, is getting the number of 'infected' down as this 'will' reduce the number of mutations. This would only be correct if they were reducing the number of the 'infected' really quite rapidly. which they are not.
So what the vaccinations, and other measures, are doing is introducing evolutionary pressure on the virus/human system to survive and keep replicating.
There would appear to be no way around this, other than by extremely rapid vaccination, probably of strictly isolated pockets.
But they aren't mentioned that, it's just get the numbers of the 'infected' down as this will reduce the pool that the virus has to mutate in - which for an organism that, if it finds a single host, can produce billions of copies of itself, some of which will likely be mutants, is IMO nonsensical, and unrealistically simplistic, as an only approach to control.
Unless they get the speed up, vastly, or enforce strict isolation, or both, then......we may be doing this dance for a 'while'.