In my opinion, Aikido works best in a chaotic combat or street situation. Not in a controlled one on one fight situation. The are a few reasons for this. I have found that Aikido works best with a committed attack, not jabbing or dancing around, but committing to a punch, strike, or grab, and it works best when you can blend instantly and they don't know what's happening.
The one time I have seen Aikido in real life was when I was in the Navy. I was young, a bunch of us were at a party, and some dude was messing with another guys girl. Guy shows up, pissed off, and pushes his girlfriend out of the way and starts to throw a punch at this other guy, bam, guy throwing the punch flies into the fall and lands on his head...fight over. He was out cold, whether from hitting the wall, landing on his head...or simply passing out (he was drunk), I don't know. At the time, I didn't have a clue as to what happened, but now I know that the guy being punched used a kaitenage technique and it happened fast, in about half a second...it didn't look pretty like in demo videos but it was very effective.
However, this is different than fighting in a controlled environment in a one on one situation where your opponent is well trained, and won't easily commit to an attack. Additionally, he understands what you are trying to do and will resist (which opens up other possibilities).
I think we don't see much Aikido because of this. Additionally, to get to a level where you can comfortable execute Aikido techniques against a very skilled opponent at high speeds takes a long time. At least a decade. For an MMA practitioner, it would be a better use of that time to train in boxing, BJJ, Judo, etc.
These are just my suppositions.