Good question and discussion. Neither Okinawan Sumo nor tegumi are present in 98% of the "styles' of karate out there. SOME aspects of it are present in Okinawan Karate hojo undo and partner drills. All out freestyle-type fights like you would see in an Okinawan village's tegumi matches are unheard of outside Okinawa or places where a significant Okinawan population exists or existed. Argentina and Hawaii come to mind. Even so-called tuite is a derivation of Chin-na which included simple takedowns, pins, holds and "pressure point" tactics. Even arts such as JJJ were derivative of indigenous fighting styles mixed with the more scientific and advanced approach of "Chuan Fa/Kempo" brought by Chin Gempin to Japan centuries ago. Yet the styles of Judo, Aikido, JJJ and GJJ/BJJ which exist today are far removed from the progenitor grappling concepts of Gempin's "Kenpo".
So what the heck am I getting at? If you learn almost anything called "MAs" these days, especially the TMAs as we know them now, you're probably fooling yourself into believing that in any way it will make you a nearly invincible fighter whether in the ring or out. Being able to throw a good punch would be a lot to ask IMO.
Even if you immerse yourself in MMAs training there is a good chance that the only time you'll use it for "real" is in competition and even then maybe not beyond sparring unless you want to make a career of it.
There are pluses and minuses to each kind of MA training:
If you are a young kid with lots of energy, lacking in self-confidence and unaware of how to use your body in SD then a GOOD TMAs school (~5% of them out there) will help one to realize some positive change by training diligently.
MMAs training will teach you strong fundamentals and of course someone well-versed in MMAs or GJJ will be able to handle themselves in a street confrontation, just as long as all the variables that could occur don't. Like gettin' "stole" on the ground by someone lurking out of sight, or the possibility that your opponent is "strapped". All things being equal, mano a mano, MMAs is fine. It will work unless you meet a streetfighter like Quinton Jackson. I have known countless cats like that, that I would pit against even a guy like Cro Cop or Chuck Liddell.
The thing is with TMAs you are going to have a very hard time finding one that will teach you anything that is practical/applicable to the real world. You have to know what is relevant before you train. Your parents have to know what is a good fighting style versus a fun time (or daycare) if you do it as a child. It is a crap shoot for sure.
With MMAs you learn some real world skills BUT how often will you REALLY fight in the street? Will you actually kick your arse too much to be able to defend yourself or will you be too old to use the skills you knew as a young strapping adult during your latter years, a time when it may be crucial for you to be young and able in body and mind albeit chronologically "old"?
How many folks want to take actual punishment in order to insure that the sado-masochism will translate to something they will only theoretically ever use? The truth is some people love to fight. It is an esteem issue whether an inferiority or superiority complex. It is overkill for the common person.
So you have to just wonder.
What the MMA guys ARE slowly finding out is that a lot of good karate techs, like spinning backfists, spinning back kicks, front kicks, roundhouse kicks, elbows, knee strikes, punches, hammer fists, basic trips, sweeps, throws, etc, are very translatable to real fighting. In fact these techs are seen in other systems like Muay Thai, Savate and Gung Fu. So it can be relevant for real fighting if taught the right way and assimilated by the right person.
As for double-legs they are in the kata and drills of good Japanese and Okinawan Karate. Patsai Dai comes to mind as does Chinto and Kusanku. If your instructor is a McDojoist you'd never know this but if your instructor can "squab"" and learned from real people then trust me you'll learn a lot of what is present in many modern systems of "MAs", plus a few "hidden truths"
Seeking is the answer. Comparing and contrasting in a relevant manner is crucial. Just like in the old days on Okinawa, advanced students should be encouraged to branch out or enhance their knowledge by learning from all the relevant sources available. Too often folks will claim that they are advanced in a system, say like Shito Ryu karate, but in fact their instructor, dojo and ryu were insufficiant to meet their (or anyones) needs. A BB often means squat in this situation.
I did "MMAs" even before the craze and I can tell you that the TMAs stuff I learned was only REAFFIRMED and VALIDATED after doing these other Mixed martial or combat-like "arts". I guess I'm just a lucky bastidge!!!
Good luck figuring it out. Peace...