No, and look at 2-24-9, which defines an illegal kick. First touching of a kick refers to the status of the ball when it is touched and not the action which caused the ball to be downfield. If there is an illegal kick taken from behind the neutral zone, then it will retain whatever status it had before it was kicked. In the case of a ball rolling on the ground, it is a fumble, and a fumble cannot be first touched. A fumble, even if it is illegally kicked, remains a fumble.
Team K punts the ball, it is muffed by B beyond the LOS, rolls back behind the LOS, and K, not knowing the rules, kicks the loose ball off the ground. Flag for an illegal kick, but what happens when the ball touches K downfield? You have an illegal kick, but because the ball retains the status it had before the illegal kick, it's still a kick, and first touching rules would still apply.
FG attempt, kick is blocked behind the LOS by B, in a panic, K kicks the ball off the ground downfield. The status of the ball was a kick, so it remains a kick, and first touching rules apply.
The change made to the status of an illegal kick a few years back made this possible. Prior to the change, the rule stated (incorrectly), that the status of an illegal kick was a fumble. In certain circumstances, that made no sense (imagine a defensive back doing a bicycle kick on a forward pass - does it suddenly become a fumble, to be recovered by anyone while lying on the ground?) But since the rule was corrected to say that the ball retains the status it had prior to the illegal kick, when you illegally kick a kick, and it's still a kick. and first touching rules apply to kicks.