How can something that came from China, to Okinawa, to Japan, then to Korea be 100% Japanese?
The term "Japanese karate" has a meaning. It doesn't mean it was wholly native to Japan--it means what was taught and practiced there.
I don't think he asked the question that people think he asked. From China to Okinawa to Japan: you are saying that the term has meaning because it was taught and practiced there, but he didn't ask how something that went from China to Okinawa to Japan can be called Japanese; what he's asking is how it can be called 100% Japanese after it then went to Korea and was taught and practiced in Korea, not in Japan.
Modern taekwondo is not kwan era taekwondo, nor is it taekwondo as seen in the sixties. The forms are different, regardless of which federation you are looking at, and in the case of the KKW forms, any Japanese kata have been replaced twice; first by Palgwe pumse and then by the Taegeuk pumse. Tournament rules were established that reflect Korean preference for kicking, and Korean terminology and philosophical perspectives were infused.
Modern taekwondo most certainly has Korean roots, but it also has roots in Japan, Okinawa and China. As for the Korean roots, most are postwar. Which shouldn't raise any eyebrows, as it is a postwar art. There are kwan founders who claimed knowledge of Taekkyeon. Personally, I'm willing to take them at face value; I know how to play baseball, football, and basketball, but I have no pedigree. If the US were occupied and a foreign culture imposed upon it and football were made illegal, after the occupation ended, I could teach youngsters how to play football, but there would be no record of me ever having played on a professional or amateur team. I simply know how to play it because it is what we played for enjoyment.
That said, Taekwondo, modern or otherwise, is not taekkyeon, and has no direct root to any specific prewar KMA. It is a postwar Korean art whose development reflects the changes and the state of Korea at the time that it was systematized. While there are apparently a few people who have links to Taekkyeon and who teach it as Taekkyeon, and while there may be unarmed ancient KMA preserved in sources such as the Muyedobotongji that people may have or may yet reconstruct, Taekwondo is not such an art. It is the KMA of modern Korea.
It is Korean inasmuch as NFL football is American football and not Rugby football.