For me elbows represent an extension of your power and balance structure. If you check out the DRA and aikido budo stuff, shite uses structures that break the connection between uke's arm and his torso. This is often a kuzushi movement because in breaking the connection you destroy uke's balance structure and can seize it. That tells me there is something to maintaining your body's structure by maintaining a connection between your elbow and your torso.
I think the connection is a bridge of transference. I think something that we struggle with in aikido is successfully transferring power into our partner. Our elbows are a bridge of transferring the power from our torso into our hands. I do not recall having seen or heard any of the gooey IP stuff yet hit elbow control in aikido. I can recall several instructors who have always advocated for breaking the connection between the elbow and the torso by extending the elbow beyond its effective range of motion.
I do not think that elbow power is simply sticking your elbow in uke's face. I know far too many good fighters that have no problem with that movement. I think it is more about aligning your shoulder and elbow (and hand) with your body structure to allow your body to accept pressure from the arm and deliver pressure from the torso. I think sayo undo is [more] representative of the internal structure of aligning your body to accept and transfer energy from the ground into your arms, while keeping focus on unified movement between your elbows and your torso. That said, I think sayo undo is not performed the same as it was when it came from aiki budo.
Finally, I think that these discussions are more beneficial when we conduct them under some modesty. As Dan puts it, most us us do not "use our center for everything", nor can it be the answer to every question in aikido - Seriously, we throw that phrase around like the Smurfs use "smurf". This is a great discussion about a significant aspect of our body posture. Yet for all our talk, I have yet to read a conclusive, compelling argument as to what is elbow power. I got my eye on this thread and I am looking forward to some smurfy responses.
Generally, the power from the ground goes up the legs to hips and dantien and then the shortest path to the point of application. If you try to route the ground power up over the shoulders, you wind up with "normal" strength, not internal strength. Some Chinese styles (particularly southern styles like Wing Chun, Hakka, etc) keep the elbows and forearmsn in that line from the waist in order to maximize the power flow. Here's a translation from one of Tung Ying Chieh's books from the 1940's mentioning the idea:
"To loosen the shoulders and drop the elbows means not to concentrate the
force at the back of the shoulders. Actually, the strength is transmitted
through the upper part of the forearm."
The idea is a fairly common one in a number of Chinese martial-arts.