Monday night, while doing light kumite (sparring) in the dojo, my opponent threw a thrust kick at me. Without realizing I was doing it, I reacted by stepping back into a neko achi-dachi (cat stance), accepting his incoming leg under my right shoulder, braced and locked in by my right arm, and raised and extended my left elbow into his chest.
This is directly out of one of our kata, and it's intended (overtly) to be a defense against a bear hug from the rear. The right arm I used to capture his leg and pull him into me was intended to be an elbow strike to the rear. Instead, it off-balanced my opponent and brought him into me. My stance was stable as I had my knees bent and dropping my center pulled him in towards my waiting left elbow strike to his chest.
You can see an example of this movement here, at 8:24
Following the elbow strike, I dropped the heel of my leading foot and transferred my weight forward, still holding his captured leg. He had to hop backwards, at which point I stepped behind his remaining planted leg and tripped him.
I did not think "I am going to use a move from kata now," nor did I think "this move from one of my kata can be repurposed thusly." I just did it, and only realized later what I had done. Would I have done it naturally without having learned the move in kata? No.
If you believe kata does not help with self-defense, you are correct. YOUR self-defense. It certainly informs mine.