I cannot speak to what goes on in the SKIF or the ASAI, or Japanese/Okinawan federations in general. What I can comment on is my personal experience. The art that I train is part of the evolution/transition between Tang Soo Do and Tae Kwon Do. While we use the term Tae Kwon Do, it could easily be argued that a better term is just Korean Karate. I remember when I first started training, that was the sign that hung on the front of the main school's building, "Kang's Korean Karate."
I practice the old Tang Soo Do/Karate forms, so my curriculum isn't analogous to the Kukkiwon or International Tae Kwon-Do Federation (the two most widely known TKD organizations). While there are many splinter Tang Soo Do organizations, most of the original kwans supported unification, so there weren't really any "official" governing bodies for the old ways.
During those transitional times, many instructors immigrated across the globe spreading what they were experts in. In my region of the USA, there were many tang soo do/ tae kwon do schools with significant pedigree of training that didn't belong to the KKW or ITF. As they passed on their art/schools to the next generation, some conformed to the new TKD curriculum, which now looks very different than what their teachers passed on to them.
I admit it's not an ideal situation, because a good governing body is beneficial. My Kwan Jang Nim joined with 5 other seniors of the old kwans to create a non profit organization in an attempt to preserve the training methods and curriculum of that era. Unfortunately, due to a number of events, it has diminished in its presence.
I know there have been instances in the past where folks with similar backgrounds to mine have been "grandfathered" in to the KKW, but since I don't use their terms, forms, uniforms, etc.. . what is the appeal? what is the benefit?