> And the ancient historian’s own account mentions another Spartan king who was unusually short and “impaired in his legs” but still a good leader.

The Spartan kings (supposedly) come from two lines of descendants of Heracles, Eurysthenes and Procles, who are said to have come to and conquered Sparta (the Spartans did not think they were native to the land). To kill one at birth would probably be a laughable idea to a Spartan.