While compounding would help, set beside modern tread configurations, the K81's tread pattern forecloses it's potential for track use. Too much light and air between tread blocks such that their edges lack support and, when heeled over, there is just too little total contact patch surface area.
In the 70's, I was production racing on K81s when the Michelins came along and they just swept the K81s aside. The K81s' semi-"Trigonic" shape looked faster but their open tread pattern gave up too much in surface area contact patch while the new French compounds coupled with their radiused rounder dumb shape somehow just stuck better. There was even a moment in time when we ordered K81s that were made in France and supposedly had French/Michelin"ish" compounding. While they were better, they still overheated as compared to those slow looking Michelins. I thought that it was mainly compound as the witness marks at the edge of the Michelins showed a narrower contact patch when heeled over. The Michelins' scrubbed contact patch had an entirely different look. It was big, a paradigm shift, the Michelins demanded more warm up, far different pressures, and a different kind of sensitivity; they lacked the Dunlop's signals and just "went," albeit at a higher point of traction loss, until you recalibrated and discovered their signals.
Looking at the K81s, I see an evolution from the K70. The K70 came from an era when there were still a lot of dirt roads, thus it harkens back to a time when many bikes were on trials type tires. Similarly, the K81, with those dentile edged strips and the air between its outside tread blocks, compared to modern roundish, interconnected tread block semi-slick patterns, still offers superior traction on a dirt/dirty road.
Getting back to the original post, don't know about KR124s but I love the non-DOT Avons.