The Lvl. 70 Summoner quest reveals that Egi actually have nothing to do with Eikons and everything to do with what happens to the aether of something when it dies. The Allagans just figured out how apply that knowledge and they don't do things by halves.

When something dies, it's aether gets released back into the environment and ultimately the Lifestream. This includes the aether of the other living things around it. If someone is really, really good at sensing aether, they can figure out what part of their own aether really started out as the aether of something else and what shape/geometry that aether has. Then they can essentially program some of their own aether take the shape/geometry of that other aether almost like it's an aetherial robot. That "robot" is the Egi.

What does this have to do with Eikons? Well people are really, really bad at calculating geometry and sensing aether (keep in mind this is the Allagans). It's really hard to tell you're own aether apart from the aether of other beings like you and the amount of "processing power" needed to calculate the geometry of living things is more then most people can handle. The Allagans did manage to make an Egi of someone specific, but that required a supercomputer powered by Summoner Soul Stones to get all the calculations right.

Eikons on the other hand, aren't living things. It's very obvious what their aether is like and their aether shape/geometry is much easier to calculate then other living things (probably comes with ultimately being based on an idea/concept). So doing all that math is actually possible. Personally, this is what I think the Ultima Warrior in Fractal Continuum (Hard) is an evolution of (knows the math of primal mechanics), but that's just headcanon at this point.

In addition, some of the geometry/shapes the Eikons have are so powerful that making an Egi out of them would require all the summoner's aether. So instead of spending aether to make an Egi, the summoner attunes their own aether to the same shape/geometry of the Eikons and gets to use their abilities, but that attunement doesn't last very long. The WoL ends up doing this with Bahamut's aether in Dreadwyrm Trance. As for Demi-Bahamut... we've got nothing. There's no lore explaining what it is or why it's called a Demi (means half/partial) instead of being called an Egi (means "essence of" in Allagan). I really hope we learn what's going on with it in the next lore book or the next expansion and Dreadwrym Trance got a whole lot of set-up.

So yeah, Summoners aren't so much summoners of primal aether as they are mystical computer programmers who steal code from their enemies.