Its origin is different from what you suspect:
The saying derives from one of Aesop
's fables, where the term is actually defined as the complete amount (all of it).
In the fable, a lion, fox, jackal and wolf go hunting, successfully killing a deer. It is divided into four parts with the lion taking the first quarter because he is king of the beasts, the second quarter because he is the arbiter of which animals get what portions of the deer, the third quarter because of his help in catching the deer, and the fourth quarter for his superior strength. Morever,
In some variants of the fable, the lion only takes three-quarters of the deer and lets the other animals fight over the remaining quarter.
This expression alludes to Aesop's fable about a lion, who got all of a kill because its fellow hunters, an ass, fox, and wolf, were afraid to claim their share. [Late 1700s] From lion's share - Definitions from Dictionary.com
The largest part or share, esp. a disproportionate portion: The eldest son received the lion's share of the estate.
A disproportionately large segment of the whole: “Though we always divided our winnings, somehow Barton always seemed to end up with the lion's share.”
Does that help?