Perhaps viruses are simply more specialized, so it's easier to eliminate anywhere to hide by innocculating affected populations, while bacteria are more general purpose and can survive even without living hosts.
If we're talking only human viruses, the count is 1: Smallpox, and it took about 20 years. If not, I think the count is 2, the other being a cat one.
In both cases, if I remember right, the virus couldn't jump species, while SARS-CoV-2 already has multiple times.
[1] https://asm.org/Articles/2020/March/Disease-Eradication-What...
As for your second point - mutations are happening whether or not we vaccinate. Vaccination does not cause mutation.
Antibiotics do not treat viruses. They treat bacterial infections, which WILL mutate and evolve as you claim with overuse of antibiotics.
This does not apply in the slightest to VIRAL infections.
Unless you can eradicate the virus or bacteria completely (mass effective vacination), the tools we create to fight them are really only delaying the inevitable.
>If some variants do evolve that escape the vaccine, odds are good that they will be less dangerous than the original, not more.
From my understanding, this is not the case. It's pure chance whether a mutation is advantageous or disadvantageous. If you are looking on a per-mutation basis, then sure, it's likely the mutation is disadvantageous. But at a population level, it's likely that atleast one advantageous mutation will emerge reasonably quickly when we are talking about something at the scale of several thousands of infections.