That's just not true, while it's true that Karate did help form modern kickboxing which is heavily incorporated into MMA but kickboxing when it was just Karate guys fighting harder really didn't quite work up until they started incorporating elements of Boxing and right now is really rather far removed from it's roots in Karate, as per TKD it really hasn't influenced MMA and modern combat sports that much, and the fighters with a background in traditional martial arts such as Karate or TKD really aren't that many and they didn't just add wrestling on top of that, they had to completely rework their striking too, learning hkw to fight in the cage from Kickboxing, Muay Thai and Boxing before they could even begin to incorporate their traditional moves in to their fighting style,

That being said I totally agree with everything else you said, I just find the part where you said

Professional fighters started off with those material arts, they just later diversified into MMA to ensure they had a wrestling base.

To be rather misleading, perhaps it's do to the language barrier, as forgive me if I'm being rude, but I take it that English is not your first language?