Earlier quoted context omitted.
There is no way to eradicate covid everywhere, simultaneously or otherwise. It's now widely accepted that the virus continues to circulate even in vaccinated people. Besides, you are not even going to vaccinate 100% of any western country, let alone all of them or the entire world. Milder mutation are what is actually going to make the virus endemic, like it happened to the Spanish flu. Reports from the South African…
There do exist viruses that have been virtually eradicated with vaccines, even if they weren't 100% effective (cf Polio). We probably won't eradicate Corona with the first gen vaccines, but that isn't the point anyway. The point ist that we should reduce it enough so that mutations are less likely.
Also, there is solid literature on the fact that vaccines during an epidemic cause stress that induces the virus to mutate. https://www.voiceforscienceandsolidarity.org