Vaccination means you don't need to limit exposure so much, which reduces selection for casual transmissibility. It also means the existing un-mutated virus has far less of an opportunity to "experiment" and mutate further.
> It also means the existing un-mutated virus has far less of an opportunity to "experiment" and mutate further.
You seem to be ignoring the time variable. If wide-spread vaccination slows the spread, then the spread lasts longer, unless and until the virus is totally eradicated.
Vaccinate enough and the virus can be totally eradicated even if the vaccine is only 50% effective. With a 90+% effective vaccine it's fairly easy. We did this with smallpox.
