A Doozy of a Rule Question

[UMP25]

Here's one I want to share with the masses here. I'm curious to see what the line of thinking on this is from people here.

R3 on third with two outs and a 2-2 count on the batter. The next pitch is a wild pitch on which the batter swings for a strike three. As he takes off safely to first on the third strike not caught, R3 scampers home. The defensive manager quickly comes out to appeal that the batter-runner was the improper batter and that the offense has batted out of order. The plate umpire confirms this, declares out the guy who should have batted, removes the B-R from first, and...?

Here's the question for you: do you count R3's run?

I've had some serious discussions with people on this, and literally, the answers I've received are split 50/50. Half the guys say no run because the final out of the inning was failure of a [proper] batter to reach first base. The other half says the run counts because R3 scored on a wild pitch independent of the batter's actions, and that this run didn't actually occur on the "play where the batter failed to reach first."

I'll share my thoughts later. I'm simply curious to see what you guys think on this interesting play.