> Syphilis is a bacterial infection
"Herd immunity" applies to bacterial infections as it does to viral. All it means is that the R-value is low enough that the disease doesn't spread. https://en.wikipedia.org/wiki/Herd_immunity#Theoretical_basi...
As an example, pertussis is a bacterial infection which most US communities have herd immunity to thanks to the pertussis vaccine.
> You can get it more than once.
There are also viral infections you can get more than once. In fact, the South African variant of COVID is likely possible to get even if you've had COVID before.
> Mixture of natural and vaccine-induced immunity.
Sure, just like nearly every virus that has been eliminated via vaccine. As, hopefully, COVID will be. Smallpox was not eradicated without a vaccine.
> We have not eradicated polio
The vast majority of the world has indeed eradicated polio via vaccination, as made clear by the fact that only three countries out of 195 still have a polio problem. The same is true for measles, rubella, and a variety of other diseases, none of which would have been eradicated from most populations without vaccines.
The OP's stance that COVID would be somehow unique amongst viruses or diseases if it needed a vaccine for eradication ignores the history of every disease for which we've mass-vaccinated against in the 20th century, as well as the vast majority of diseases that have not been and never will be eradicated by herd-immunity-reached-through-mass-infection.