Perhaps we're not on the same page as I have a different definition. Kata is an excellent additon to an art, particularly if proper bunkai is taught, however, many arts don't use this concept of teaching. I would not define their training as lacking because of it. Sparring can also be a fine addition, but only within the context that it is taught i.e. sport sparring is a poor choice for self-defense and vice-versa.
When I think of complete, I'm thinking within the context of the arts purpose. Perhaps from a sport or hobbyest (which is fine) perspective, KKW TKD can be considered complete. I view martial arts more from a self defense perspective so I always think of the art being viable at different combat ranges, ground defense, different options other than striking i.e. locking/throwing/pressing etc, weapons, pre-fight tactics, different clothing, different settings and environmental stimuli etc.
As I've mentioned in the past, perhaps that is one of TKD's strong suits in that it has different aspects for different goals. And perhaps that can also be considered part of being complete. Hmmm..'part' of being 'complete' sounds odd
The most obvious and prevalent self defense view that I don't agree with is that you must train every single scenario for competent self defense. I don't bother training in the dark, I don't bother with different sparring rulesets, I don't bother with street clothes. I tend to put faith in a practitioner's understanding of the principles in Taekwondo/karate/jujutsu/whatever to deal with different situations and apply them as appropriate, rather than having to be shown "this is what you do in this exact situation". There isn't a better tool to teach this than full contact sparring, and I'm of the opinion that well developed striking from taekwondo, karate, kickboxing people will serve them just fine without throwing, submissions, etc. just like well developed throwing or grappling serves wrestlers and jiu jitsu people.