Viruses and bacteria don’t necessarily “go extinct” (get completely eradicated from the face of the earth) but their harms can go extinct. That’s because immunity to pathogens is more important than exposure to pathogens — your immune system gets “the final say” over whether they can harm you.
Prior to smallpox vaccination, the death rate due to smallpox was coming down in Boston:
Using extrapolation, even though it is statistically haphazard to do so, then by 1920 in this graph, the death rate would have hit a “zero point” — indicating no more deaths from smallpox from then, onward (and maybe, forever). That’s without a smallpox vaccine.
Here are data on the percentage of all deaths in London due to smallpox:
There is a decrease in the share of all deaths due to smallpox beginning almost 40 years prior to the advent of a smallpox vaccine. And here is admittedly-haphazard extrapolation from the 35 years prior to the advent of the smallpox vaccine:
Of note is that the actual death share in 1902 was 1.6%, and the projected death share — with no smallpox vaccine — in 1902 was under 2.0%.
Here is the yearly smallpox death rate per 1,000 in Sweden prior to smallpox vaccine:
While still haphazard, the projected “zero point” in smallpox fatality — in the complete absence of a smallpox vaccine — would have been by 1880.
Evidence strongly suggests harm from smallpox was decreasing prior to the vaccine and that — in the complete absence of a smallpox vaccine — evidence very weakly suggests that human harms due to smallpox would have been entirely eradicated by 1930. This eradication, if it would have happened, would be due to built-up immunity.
Smallpox would still “exist” in some shape or form, but smallpox “deaths” would not.
Reference
[smallpox in Boston] — https://sitn.hms.harvard.edu/flash/special-edition-on-infectious-disease/2014/the-fight-over-inoculation-during-the-1721-boston-smallpox-epidemic/
[London and Sweden smallpox trends] — OWID. https://ourworldindata.org/smallpox