No, this is wrong. Had a high enough percentage of the population been vaccinated it would have helped to prevent the spread of the disease, including the variants. Places where a low percentage of the population have been vaccinated are the areas where Delta is spreading rapidly.
Oh, and the last I knew studies have shown that any immunity to COVID from having contracted it before fades in time. It is recommended that those who have had COVID should still be vaccinated. My two oldest sons contracted it last year, both received their vaccinations as soon as they were able. They are both COVID long haulers (Google this if you don't know what it means), and wanted no part of catching it again.