The relationship between Tang Soo Do and Shotokan is very clear.
First for approximately 50 years before WWII, Korea was conquered by the Japanese, and very horribly abused in the process. Any natural Korean arts were surpressed.
Judo (Korean Yudo) came to Korea and after Funakoshi Ginchin (and a group of other Okinawan Instructors) shared Okinawan Karate in Japan, especially as Shotokan was taught at the Naval War College, it is very likely that it was the Military that were practicing Shotokan in Korea and the likely point of insertion.
Besides the relationship between Tang Soo Do and Karate as terms, the Kata curricula fo Tang Soo Do Moo Duk Kwan is almost identical to Shotokan (abet with some modifications), and most of the kata have names which are also very similar.
Not just TSD, but TKD, when General Choi first published a text on Tae Kwon Do he included all of the Shotokan forms in it along with his new forms (and if you ever see a copy you can see the similarities).
Likewise Hapikido (one group of which) goes back to a Korean training in Daito Ryu Jujutsu.
At the same time, since the war you will not find Korean references of these origins. After all if the Japanese had brutally used your country, your women and your resources for generations why would you want to acknowledge anything you got from them.
Nor should they. That doesn' t change origins, just makes it very muddy water to discuss.
When TKD was in the Seoul Olympics, Japanese competitors faced incredibly hostile Korean audiences. The past isn't forgotten, as much as the Japanese keep pretending it didn't exist.
So what is the truth about original Korean arts, very, very, very hard to determine. But then much of the past of all arts will remain muddy, too.
Pleasantly,
Victor Smith
Bushi No Te Isshinryu