Or is it impossible for any lava to cool into rocks with that range of silica?

Not impossible, but not common. The two most common lava types are indeed mafic (mostly basalts) and felsic (mostly rhyolite). A well known intermediate lava type is andesite.

Why?

This question of why most lavas are either mafic or felsic has been bothering geologists for a very long time. This problem even has a name: the "Daly gap", named after the geologist who first identified the problem almost 100 years ago. There is no easy answer, and it's still being debated. A good (yet somewhat technical) read is available here in this ResearchGate question and following answers.