I would actually argue that a Karateka can gain a great deal of benefit from adding Bjj and/or Judo to their skillset. The main thing those arts will add is the removal of the disadvantage a karateka deals with when someone passes their striking range. This is something I dealt with when I did karate, and its something I still see when I practice with karate exponents; Once you're inside their punching and kicking zone, there's really no mechanism to deal with that opponent except possibly knees and elbows, which frankly aren't as well developed in karate as they are in Muay Thai, and even Muay Thai practitioners have problems dealing with it. Speaking of Muay Thai, another option for karate practitioners would be adopting the Muay Thai clinch. Beyond that, I completely agree with the term "Seizing" for what we see in Karate, maybe Aikido as well.