Do people say “文学家？” Do people use "文学家” normally？ It feels like it could be a correct  word, but no one would say "literarist" on the other hand in english.
This is 文学家 is a "big" term. You call Shakespeare and Mark Twain 文学家, because not only did they create literary works or treatises, they also made deep influences in language and literature. You use 文学家 because you want to emphasize the latter part. This word is reserved to the well achieved. 

By comparison, 作家 is a more common term. In fact, anyone with any literary work published could call himself a 作家. 

This is why you get 2.5M hits for 文学家, versus 65M for 作家 in google.

Example of usage:


Discussing our favorite authors: 作家.
Talking about Shakespeare's literary achievements: 文学家.
Talking about one of Shakespeare's work: 作家 or 诗人.
Talking about George R. R. Martin: 作家. Sorry George, you might want to wait until you're dead.


So people do use 文学家 normally, but not in the sense that it is an ubiquitous term.