Runner is on 3rd base. Batter in the box hits a foul ball on the ground down (so now it's a dead ball) the first base line. The first base coach bends down and tosses it to the pitcher. The pitcher misses the catch and it rolls over to 3rd base. The runner steps off the bag and flips the ball to the pitcher, and the umpire calls him out. Is this the correct call? (for the sake of this example, I won't mention the umpire was standing near 3rd and said "hey! a little help." and that's why the runner grabbed the ball). This doesn't appear to be interference as the ball is not in play, and is dead.
A runner can be put out directly via any of the situations in MLB Rules 5.09(b), or on appeal via any of the situations in 5.09(c). None of these cover a runner touching or being touched by a dead ball.
As you mention, 5.09(b)(3) (interference) shouldn't be relevant because the runner is not interfering with a play.
That said, the offensive team (especially at the professional levels) should be primed to not touch the ball just to avoid odd situations. Let the defense deal with the ball.
No play is being made, the interpretation from the description is that the runner was trying to help speed up play and not create problems for the defense, there's nothing mentioned that comes close to a reason for the runner to be out.