Unfortunately, I had an interesting lesson in overdoing a quick-turn at Barber the last day of CSS that I think is relevant to this discussion. Unfortunately it resulted in a crash.
I was riding hard attempting to catch up to my coach who was chasing another student and had the right side of my tires nice and warm as a result. However, the extreme left edge of my tire was still relatively cold (another of the many lessons from this crash) and I wasn't aware of that. Coming into the final (left) corner prior to the front straight, I quick flicked the bike in at a very high speed (both the flick and the entry speed) and almost immediately lost the front. Fortunately, nobody was hurt and damage was minimal.
The quick-turn lesson from this, after much discussion with several coaches, was: the fast rate of steering didn't DIRECTLY cause the crash, it was the cold edge of the tire once I got it there. The tire was able to take the FORCE of the quick-steering just fine, but the reason the quick-flick was an issue was that I didn't get the chance to feel indications that the tire edge was cold in time to stop leaning the bike over. Had I steered the bike more slowly, I would have been able to feel the cues the cold edge would have given me in time to prevent the complete loss of traction. The bigger issue was really my awareness of the cold edge on the less used side of the tire, but since this thread is about quick-turning, I thought the quick-turn lesson would be a good one. This is a big reason you don't quick-turn on cold tires... It's less because they won't take the force while steering, and more about giving yourself a chance to feel where the cold tire traction limit is before you exceed it.