“However, as vaccine charity GAVI explains, viruses are mutating all the time”
Yup, kind of like the climate, which has ALWAYS and will ALWAYS be changing!
As to virus mutations, my understanding is that they are usually in the direction of becoming more contagious but less lethal.
1) "Usually" the mutations are insignificant. However, you are correct that of the small percentage of mutations that are significant mutations affecting contagiousness and lethality, most are in the direction of more contagious but less lethal. A much smaller number will be more lethal. Generally, for a variety of reasons that don't even involve, say, viability of the virus with time outside the host, greater lethality promotes LESS spread of a pathogen, not greater spread.
That said, given enough chances, more lethal strains do pop up from time to time, because there are so many chances for a mutation to occur: A contagious person can output 200 million virions in a single sneeze. An infected person themselves can carry 10^9 to 10^11 virions. That is a lot of chances for mutation!
My own guess is that usually mild diseases may have a sort of genetic momentum that keeps them mild but successful, success being very effective spread.
2) Notice what just popped out above, though. Of "significant mutations affecting contagiousness and lethality, most are in the direction of more contagious but less lethal." This is very well known, studied, and documented. But from the pathogen's point of view, that IS success. So, the idea that no mutations are beneficial to any mutated organisms is complete hogwash.