A rational agent seeks to maximize utility. It turns out that murdering people for their organs does not maximize utility! At the very least, society is cleverly structured in a way that the probable utility of murdering somebody for their organs is relatively low.
Basically, my point is that your argument is a straw man. The issue isn't too much logic but rather than you (the agent killing everybody in the example) did not use enough of it. For some reason, all the "arguments" about the inferiority of logic always assume the logical person is extremely short-sighted or ill-informed.
And I don't think short-sightedness or ill-informing was ever an integral part with arguing about how empathy is essential to human live. IIRC there was an example of perfectly informed and rational computer of super human intelligence who was told to calculate a particularly difficult equation, to which he promptly restructured all matter on earth, including people into a giant super-computer.
As a hacker without empathic skills I could pick any goal and ruthlessly implement it: Goal: Need more organ donors 1. People are unwilling to donate organs 2. Force all prisoners on death row to give away they organs for staying in death row accommodation 3. Kill them in ways that doesn't hurt the organs (they might suffer some inconvenience) 4. Harvest organs
As Charlie Chaplin in Great Dictator said: "More than machinery we need humanity, more than cleverness we need kindness and gentleness, without these qualities life would be violent and all would be lost."