Rather more good MMA fighters come from muay thai or BJJ or wrestling backgrounds than they do TKD, so I am not convinced sport TKD is a good entry path into MMA. By the way, I don't necessarily equate MMA with self-defense training either, although for sure, it's a huge leap up in practical training compared to most sport TKD.
As for the karate bit, that's only part of the picture. I am convinced that the Korean martial artists that were around during the formation of TKD were very well-rounded in their technical skills thanks to widespread study of yudo/judo. They had also yet to develop the focus on fanciful kicks that came later during the late sixties and early seventies. That combined with a tough-minded training atmosphere thanks to the war footing Korea was in for decades meant they produced good fighters that could be effective from all ranges.
I don't think there is anything wrong with teaching kids, but a distinction should be made. Too many lines are blurred when the adults and the kids are taught the same sanitized and sportified curriculum and then that is what is passed on as tae kwon do. That's a problem and it's a big one to fix because all sorts of nasty, unwinnable issues like child black belts, sport sparring and empty forms are bound up into the discussion.