As someone who used to participate in those karate tournaments growing up, I can tell you the main thing is not where you strike someone, but how hard. You could strike to the face with a punch or a kick, but if you hauled off and drilled someone, you'd be thrown out of the tournament. It was really an odd thing, as in you were supposed to strike each other, but still "respect" your opponent and not try and knock their head off. It was more like fencing than it was combat, where you got a point for the "strike" to the head, stomach, or ribs, but if you tried to hurt them, you would be penalized. So in that sense, Daniel walloping Johnny with a full force kick to the face would've gotten him disqualified. So I suppose Johnny was correct in that it was an illegal strike. Also, you had to wear hand and foot pads to fight in those tournaments. Not that they did anything, they were like 2 ounces.
Also, Cobra Kai is fucking awesome, and if you haven't watched it yet, you should drop whatever it is you're doing and go watch it immediately.