???While variolation was indeed developed to deal with smallpox, the fundamental principle, inoculation with a weakened strain of the disease, is applicable to a wide range of bacterial diseases.
Was variolation used for anything else? I haven't heard of it, so I don't think it was used in Europe at least.
Vaccination (for smallpox, using the related cowpox virus) was introduced by Edward Jenner in 1796; AFAIK the next major step was by Pasteur who found that weakened versions of bacteria could be used to immunize against anthrax and chicken cholera (1870s?).
And this was with (by then) a good understanding of germ theory.
OTL, a vaccine for yellow fever wasn't developed until 1837, apparently, and there are still diseases we don't have vaccines for.
So, I don't see that variolation is actually particularly extendable...