Grammar of elimination reactions
"A gets eliminated" or "the elimination of A" is just plain wrong, typical sloppy lab slang by many (too often native?) speakers. Very common, but still terrible. 

As you say, the small molecule X gets eliminated from substrate A. Chemical English is still just English, and that is the correct way to put the case in question into words in this language. 

"A undergoes elimination" (short for "A undergoes an elimination reaction) is a widely used, and imo also correct, variant.