What evolutionary advantage do viruses have in host specificity?
It is true for any living creature, that it would be great for it if it could thrive in all environments. Any creature would do better if it had a greater ecological niche while remaining as competitive in each of these niches. However, competition, predation and other biotic and abiotic factors lead species to specialize in specific niches. Of course, some species are more generalist and some are more specialist but I won't go into these details.

When it comes to parasites, such as viruses, the story is the same. A host is an environment. Being less specific would be great but the immune system is no easy detail to get around. Viruses are often quite specific to a given species, just because it evolved to be efficient for a given host but tend not to be that efficient in other hosts.

Note that parasites are not only species specific but also often tissue specific and specific to the specifics genetics of the host (e.g. malaria).

Somewhat related posts:


Why do parasites sometimes kill their hosts?
Why aren't all infections immune-system resistant?




Thank you @DeNovo for helpful comment