Great Post!!!! I have some more to think about!!!
The real question is how reductive do we wish to get? Karate most definitely stems from China. It only reached the Japanese mainland in the 1920s. Korea was occupied by Japan from 1910-1945. However, it always had trade and intellectual exchange with China. China absorbed trade and commerce from across Asia and the steppes. Ideas arrived indirectly from Europe and in limited cases directly -- Alexander the Great, Marco Polo, Jesuits, et cetera. Reasonable minds can deduce that martial arts in the broadest sense were part of these exchanges. Certainly in the Ryukyu kingdom is a famed instance of such an exchange. Scholars and elites who were trained in China brought back the foundation of what became karate. Over time karate developed unique traits. Nationalism led to the removal of Tang from the name to the simpler "open hand."
The paradigm of outside influence and the subsequent repurposing is exemplified in karate. Arguably, TKD fits this bill. Although the Koreans were occupied, their culture was not eclipsed. Any incorporation or adaptation of karate was filtered through their own lens. It has gone on to be a unique art in its own right. How much past forms of Korean fighting styles informed TKD at the inception or in a later stage of progression is probably impossible to dissect with English language sources.
In closing, ideas that are exchanged between cultures are rarely adopted wholesale. The anger between Chinese, Korean and Japanese cultures has led to overstatements of individualism and overplayed the denials of outside influence.
One more point, simultaneous, independent development of ideas has occurred throughout the world over space and time. Learning to use one's body as a weapon, maximizing on physics, has happened across the globe.