The question asks about the origins and lineage of Kuk Sool Won.
In the past, I once had an interest in KSW as well. At various times, I've looked for this information, hoping that something authoritative would appear. To date, I have not found it. My tentative conclusion is that there is no widely accepted and verifiable source of historical documentation for Kuk Sool Won which would prove its claims of origin and lineage.
That said, if you enjoy practicing KSW, that's all that really matters. Even if its history of transmission turns out to be fictitious in part or in whole, that shouldn't stop you from enjoying it for what it is.
Beyond KSW, most Korean martial arts have a tumultuous relationship with the truth with regards to their origins and lineage. I attribute this to the Japanese occupation of Korea 1910-1945. For 35 years, Korean culture and especially their practice of Korean martial arts was suppressed. For two generations, young Koreans were forced into the Japanese military, brought to Japan, and were exposed to Japanese martial arts. The biggest influences were Shotokan Karate, Judo, Kendo, and Daito Ryu Aikijujitsu. If you compare those martial arts with Taekwondo, Hapikido, and Kuk-Sool-Won, you'll see a lot of the same elements. To some degree, these elements are universal, and so you'll expect to see some similarity. But there are things that were simply copied wholesale that can not be anything but a copy.
For example, in early Taekwondo, they did the exact same forms (hyung) that Shotokan karate did. They might have Koreanized some of the names, but the forms were more or less identical. That's documented and completely verifiable. Later on, Taekwondo developed its own forms, rearranging patterns from Shotokan karate forms into new forms. Those were affectionately referred to as the "blender" forms. And that's documented as well.
Taekwondo has been evolving away from its Shotokan origins ever since, but the influence is unmistakable. Taekwondo today is still a derivative of Shotokan karate. And yet, you will scarcely read about or hear about the Japanese origin of Taekwondo from any Taekwondo source.
I have a black belt in Taekwondo, and yet I didn't know about its origin in Shotokan karate until after I looked into it myself, well after I had received my black belt. There was so much anti-Japanese sentiment in Taekwondo that I just didn't think such a connection was possible. Even the ITF Taekwondo forms were sometimes named after Koreans who killed Japanese people during the occupation. And we had to recite their history: "Joong-Gun: Named after the patriot, Ahn Joong-Gun, who assassinated Hiro Bumi Ito, the first Japanese Governor General of Korea, in the year 1909."
https://taekwondo.fandom.com/wiki/Joong-Gun
https://www.taekwondopatterns.info/joong-gun
You'll find this anti-Japanese sentiment in a lot of Korean martial arts. It's because Koreans hated the Japanese for invading and occupying them for over 35 years. The Japanese suppressed all things Korean, especially their native martial arts. After two generations passed, hardly anyone remembered Korean martial arts. Fragments, blurry memories, and concepts survived and influenced contemporary Korean martial arts.
Whenever we hear about "pure" Korean martial arts today which supposedly did not derive from Japanese martial arts, it's met with a great deal of skepticism. Usually, claims of pure Korean origin and lineage do not come with any form of verifiable documentation to confirm those claims. The Koreans will argue that such paper documents were destroyed or lost during the occupation, which sounds plausible but still leaves the question unanswered.
So nowadays, Korean martial arts don't often mention their Japanese roots. And indeed, they often are guilty of revisionist history, claiming that most of their curriculum comes from ancient Korean sources, prior to the Japanese occupation. When martial scholars have analyzed these sorts of claims, they tend to come up short. And usually they conclude the opposite, that there is almost no direct transmission from those older Korean arts to the newer ones. There is definitely some influence there, but not much direct transmission.
Kuk Sool Won was created, in part, to give Korean people their own martial art. Its name even means, "National Martial Art". Many other Korean martial arts practitioners wanted the same thing after the Japanese occupation had ended. They saw that Korean culture had been destroyed by the Japanese, and they wanted some martial art that Koreans could call their own. They absolutely wanted nothing to do with the Japanese. So you will find nothing but denial of any claims of non-Korean and especially Japanese derivation from within the style.
In conclusion, I think you should be skeptical of any claims about Kuk Sool Won's origin and derivation. But don't let that get in the way of enjoying it if it's something you want to do.
Hope that helps.
