R Dovid Ribiat answers your question in his book The 39 melochos (vol 3, p. 888)
It is questionable wether killing a mortally ill or wounded animal is
the melocho m'deoraisa of Schochait. In any case, doing so is not
actually permitted. "Mercy killing" situations are common examples of
Example: One who notices a badly wounded bird ...
On page 145 of "Call of the Torah" by Rabbi Elie Munk, he says, quoting the Sefer Divrei Yirmehayu, "Tradition tells us that the tower-builders suffered three punishments. The Semites, who wished to perpetuate the memory of the Flood's victims with a memorial, were scattered over the earth--the lesson of the Flood should have been taught to ...