The more people that have a virus, the more copies are out there mutating, and the more mutations (in total) you'll see.
If you introduce negative pressure, like a vaccine or antibiotics, some of those mutations that were previously deleterious will turn out to be a net benefit. E.g: A thicker cell wall in bacteria makes it harder to absorb nutrients, but also increases the dose of antibiotic it can survive. These mutations happen all the time by chance, but in the absence of the vaccine/antibiotics, they are less fit and will tend to die off.
A vaccine that eliminates a virus is likely to result in less infections. Less infections means less mutations means less chance for a dangerous variant. If some variants do evolve that escape the vaccine, odds are good that they will be less dangerous than the original, not more.
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.
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.
Finally, human DNA is peppered with traces of viruses.
https://www.smithsonianmag.com/science-nature/virus-genes-hu...
That isn't a real worry, that is a science-fiction worry.
Viral load, transmissibility and virulence are fairly tightly coupled and they will definitely not come fully uncoupled like that.
https://www.cambridge.org/core/journals/parasitology/article...
Anderson, R. M. and May, R. M. (1982). Coevolution of hosts and parasites. Parasitology 85(Pt 2), 411–426.
Frank, S. A. (1996). Models of parasite virulence. Quarterly Review of Biology 71, 37–78.
Alizon, S., Hurford, A., Mideo, N. and Van Baalen, M. (2009). Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future. Journal of Evolutionary Biology 22, 245–259.
It's not 100%, but it seems to be more than we hoped for. At least for covid, the viral load is reduced and infection depends on that.
Now if we have a vaccine for those, do those relatively benign viruses evolve to bypass the vaccine, but in doing so pick up more dangerous mutations in that process?