With regards to Okinawan karatedo, the link to its Chinese origins are not as obscure or hidden as it is in Japan and Korea. In fact, karate was originally written as 唐手, meaning "China Hand" (pronounced Tode/Toudi in the Okinawan dialect). The Okinawans also referred to their martial arts as kempo 拳法, the Japanese pronunciation of the characters for quanfa (another generic name often used for Chinese martial arts). The name kempo itself was used both interchangeably and together with karate in Okinawa for many years, until karatedo 空手道 became standardized on mainland Japan. Trade between China and Okinawa is well documented, as is the long-standing influence of Chinese martial arts, whether we're talking about Shuri Te (and its descendants) derived from Matsumura Sokon's study of quanfa or Naha Te (and its descendants) derived from Higashionna Kanryo's study of Bai He Quan 白鶴拳. In fact, Goju Ryu, the most prominent descendant of Naha Te, was even awarded honorary koryu status by the Dai Nippon Butoku Kai in 1998 (although the term koryu specifically refers to Japanese martial arts founded before 1868) because of its long, unbroken line of transmission and history.
Moving onto Korean arts, the most common martial arts practiced (Taekwondo and Tang Soo Do) come directly from Japanese karatedo. Tang Soo Do itself is just the Korean pronunciation of 唐手道 and is in essence simply Shotokan (with some Taijiquan influence if we're referring to Moo Duk Kwan specifically). It's only an ingrained hatred for anything Japanese (perhaps rightly so) that forces Koreans to deny this connection and claim Taekwondo is some indigenous art supported by 2,000-year-old cave drawings, when in fact Taekwondo was created solely because Tang Soo Do was "too Japanese." Even the fighting arts derived from the MooYeDoBoTongJi, a Korean military manual from the 1400s, can be traced to China and the Shaolin temple. The manual itself is pretty much a copy of an older Ming Dynasty military manual, Jixiao Xinshu, written by General Qi Jiguang. As if that wasn't conclusive enough, the MooYeDoBoTongJi contains an entire section of kwon bup, which is simply the Korean pronunciation of quanfa.
In Japan, the waters begin to get a little muddier, but why is that?