Hanzou
Grandmaster
"Grappling" is a spectrum. On one end you have wrestling, judo, and BJJ. Here, grappling is the end object, finishing the fight, whether sport or actual, in most cases on the ground with submission or points based on the grappling technique.
On the other end is karate where grappling is only the means to an end, the end mostly being a finishing strike with the victor usually on his feet (sometimes on a knee after a takedown.) Its main purpose is to set up and facilitate the strike. In kata, this is often hidden, but in application, it is of great importance (to those educated in its use.)
With such differing goals, the grappling techniques from one end of the spectrum are of limited use to the other end. Naihanchi is called a "grappling" kata (most all Okinawan kata are to some extent,) but the grappling is of a much different kind than judo or BJJ. Attempting to put a judo or BJJ interpretation into a karate form is likely doomed to failure.
I admit that when teaching kata, I use the term grappling. But it must be understood that it is much different than wrestling grappling. For this reason (especially after reading this thread) I like referring to karate as a seizing/striking art.
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.