The only thing "circular" here is some of the logic, or lack of it, being applied to the play under discussion. There appears to be some confusion as a result of the batter/runner being thrown out at first prior to the appeal being sustained on the runner from first who missed second on his way to third. In this play, the out on the batter/runner does not remove the force on the runner advancing from first to third. That runner, as soon as the batter hits the ball, is forced to advance to second. The fact that the appeal is sustained after the batter/runner is thrown out at first has no bearing on the fact that the runner on first was forced to advance by reason of the batter becoming a runner. The order of the outs sustained seems to have injected an element into the play that is confusing quite a few commenters here, but the rule itself is clear. No run can ever score on a play where the third out is the result of a force play. The runner from first was forced to advance to second. At the moment he missed it, the batter/runner had not been thrown out at first, and the fact that he was thrown out at first before the appeal on the preceding runner was sustained does not negate the fact that the runner on first was forced to advance and was called out for not reaching the base to which he was forced to advance. Overcomplicating this is creating a lot of confusion, but reducing this play down to its basic components makes it relatively simple to figure out.
So to answer the question, "Umpireplb, did you read the thread at all," my reply is yes, I did. "R1 and R3, 1 out. Batter hits single to RF." That's the situation. "R1 advances to 3B but misses 2B." That's the next part of the play. The runner on first did not touch the base to which he was forced to advance. "F8 throws the batter out at first base for the 2nd out." Two outs. "The defense properly appeals that R1 missed 2B." The key now becomes: did the play on the batter/runner remove the force on the runner at first who advanced to third? The answer is NO, it did not, because according to rule 4.09, "with two out, the runner is out at the moment he misses the bag, if an appeal is sustained as applied to the following runners." In this case, technically speaking, the runner missed the bag when there was still only one out, and the out at first occurred AFTER the runner had already missed second base. So that out at first has not removed the runner's responsibility to touch the base to which he is forced to advance by reason of the batter having become a runner. The fact that the batter/runner is called out at first before the appeal was made on the runner advancing from first to third, again, does not negate the force on that runner. Forget about the "order" of the outs on this play; the appeal is throwing a monkey wrench into an accurate interpretation of this play. The key is, was the runner on first forced to advance? Yes. Was his out at the base to which he was forced to advance the third out of the half inning? Yes. The out at first on the batter/runner has not removed the force on the runner in front of him (the runner on first) at the moment the runner missed the bag. Can a run score during a play in which the third out of a half inning is the result of a force play? No.
If anyone is still confused, try thinking of it this way. Forget the appeal. Same play, just no appeal. One out, runners on first and third. Batter hits the ball, so the runner on first is forced to advance but is thrown out at second, or even third, it doesn't matter. Two outs. The batter/runner is thrown out before he reaches first base. Three outs. Does anyone think the runner on third's run would count on this play? It definitely would not, because the third out of the half inning (the out on the batter/runner before he reached first) was the result of a force play. So in the play under discussion here, the fact that the out on the batter/runner came before the appeal at second was sustained does not make it a "second out" that removes the force on a preceding runner. That out at first is the third out, not a force-removing second out, because the runner advancing from first to third is already out, before the batter/runner, at the moment he missed the bag, not at the moment the appeal is sustained. Again, this is what the rule says: "...except that with two out the runner is out at the moment he misses the bag, if an appeal is sustained as applied to the following runners."
Does this help? I hope so. I went through Harry Wendelstedt's umpire school training five times over a period of twenty years and took those damn written tests enough times to know unequivocally that I am correct about this ruling. This is not arrogance on my part, it's simple repetition and memory, supported and amplified by thirty-two years' experience and more than six thousand games umpired. No run scores on the play.