The play opens with three people: Kent, Gloucester, and Edmund. At first I was puzzled about why Edmund is there. Kent and Gloucester, being part of the king's council, I could handle. But why Edmund, the villain? Here is my attempt at explaining Edmund's presence.
Shakespeare represents with these three a whole soul: reason, spirit, and appetite. Kent, being the king's advisor, is reason. Gloucester, the king's loyal friend, is spirit. Edmund, greedy for power, is appetite.
Kent's first words are "I thought . . ." Thinking is exactly what reason does.
In the Platonic tradition, the spirited element of the soul mediates reason and appetite. In the opening scene, Gloucester introduces Kent to Edmund.
Lear enters then and, in his madness, which is already upon him, tears apart this whole soul: he sends Gloucester to attend to the dukes of France and Burgundy. (Edmund, though not addressed by the king, leaves with Gloucester.) I say that Lear is already mad, or at least (lit.) unreasonable, because there is no good reason to disrupt a whole, well-ordered soul. A whole soul (or city) is a beautiful and good thing, and there can be no good reason (or no reason at all) to tear it apart. But Lear does tear it apart; I conclude that Lear is not governed by reason even from the very beginning of the play.
Tragedy ensues. What else would one expect to follow in the wake of such a mindless disruption?
After Lear has made the tragic decision to banish Cordelia, each of the three parts of this original triad proceed thusly. (1) Kent tries to dissuade the king from his course of action. But Lear will not listen to reason, and reason separated from spirit and appetite can affect no change. (2) Gloucester, eager to assist the king, overplays his hand since he is not guided by reason. (3) Edmund, unbridled from reason and spirit, dwells on and pursues his base passion for power.
Thus, Edmund's presence is explained by the necessity of appetite in the city/soul. Appetite is not bad; but appetite following its desires leads to ruin. Perhaps Kent and Gloucester together could have saved Edmund. Consider Gloucester's urging Kent to remember Edmund as Gloucester's honorable friend and Kent's statement to Edmund that "I must love you, and sue to know you better." Kent however is prevented from taking Edmund under his wing, and so Edmund is not tamed, as he might have been. Apart from Kent and Gloucester, Edmund ruins both Gloucester and himself. (Reason seems to survive.)
Last thought (for now): It is true that Lear's division of his kingdom among his three (or two) daughters is the obvious source from which the rest of the play flows. But his disruption of the Kent-Gloucester-Edmund tripartite city/soul allows the second division (the division of his kingdom) to proceed. I suggest that if Gloucester and Edmund had been present, they (with Kent) could have functioned as a well-ordered unit to prevent (or at least soften) the disasters brought on by Lear's decision.