Does a character (presumably the protagonist) have to be definable to be engaging? In a word, yes. The character’s nature could be defined as anything. In fact, inconsistent suggests conflicted (internal conflict), but the audience must be able to define the character in some way. Otherwise, they cannot connect or care. I’ve struggled with protagonists whom readers didn’t care about for reasons similar to what you’re facing; they were do-nothing characters.
Plot, story, and character while independent concepts, overlap. A character driven story is not devoid of plot, because a story without plot provides no opportunity to demonstrate the protagonist’s character. Often we confuse characterization with true character. Characterization are all of the details of a character - what they look like, where they’re from, how they talk, walk, or dress. Also, what they say to themselves in they’re own head. This is all surface, adding little to the story. True character is demonstrated by the decisions the protagonist makes in response to events (plot). Particularly, in response to dilemmas, the high stakes decisions. When the bulk of the plot results from the world reacting to the protagonist’s decisions, that is a character-driven story, but the character must decide.
If I follow you’re question right, your protagonist lacks empathy. His or her initial decisions must demonstrate this lack of empathy. The events which result would then suggest to the protagonist that he or she must change. Thus, the decision to take the empathy pill. Or the world could force the taking of the pill, but this would lean away from character-driven.
Readers connect with characters who feel real. To feel real they must act (make decisions). Without the choices, the character is a sketch, a police description describing the subject’s appearance - not a person.