The sacrifice of an Amalekite male

Peace to the honorable Rabbi Michi,
 
In "Moves Among the Standing," the rabbi argues that it is possible to give moral validity by providing a rational explanation for the destruction of the Amalekites. The rabbi reasons this by saying that the Amalekites may have posed a real, and even potential, danger to the Jews (p. 80).
 
On page 418, the Rabbi attributes the word "male" (" He wiped out the male of Amalek") to the physical death of Amalek.
 
So, why was it that when Saul destroyed the Amalekites, he was instructed not to spare the Amalekites' animals?
This does not fit with the two things I mentioned in the Rabbi's name above.
 
Best regards, Ehud