Do you see how this absolutely requires not just a lowered infection rate with lowered severity of infection? You need a less than ~10% chance of infection in the vaccinated/otherwise immune for the math to work. If the chances of infection after vaccination are still 80% then there is still a large reservoir of potential carriers and you don't have herd immunity.
For most of history what people expected out of a vaccine was immunity from infection not still getting infected but with less severity.