I've eavesdropped on that part of the debate where some people are insisting on shuffling the stub with the muck. Assuming that I've understood the situation correctly and there is still a portion of the original stub intact, I don't think they are right at all.
Perhaps the wording of RR-4 can be clarified, but I think the "spirit" of the rule is exemplified in (1) and (2): in cases where some cards in the stub gets mixed in with the muck, we should always try to salvage as much of the stub as possible, and to complete the hand using only those "fresh" cards and not cards in the muck. If you read all three parts of RR-4 as a whole, I think it becomes clear that (3) is intended to only apply when it is not possible to salvage enough of the original stub to complete the deal.
We have to realize that the act of reintroducing already-mucked cards into play, which at least some players at the table have already seen, is an extremely drastic measure. There's always the possibility that some information has been passed between players during play that identify already-mucked cards, and there is no good reason to give a chance for those cards that have been voluntarily folded by players to come alive again. The only reason why we'd even consider (3) is when we can't salvage a sufficient part of the original stub and we are desperate -- because the even more important principle is that any hand must be played out to completion once substantial action has occurred.
In this case, I see absolutely no reason to invoke (3) and allow mucked cards to come back to life, when a new stub can be created consisting only of fresh cards. Some people will argue, however, that if we don't shuffle in the mucked cards we also don't give the original river card a chance to come back out either. But we haven't seen the original river card. Not allowing the original river card to come out may cut down the odds of the drawing player, but it may not; it is also possible that the new river card will be helpful when it would otherwise have not come into play. Put another way, some of the times the old card would have helped the drawing player and sometimes not; sometimes the new card helps the drawing player and sometimes not -- both possibilities are equally true for the opponent as well. In truth, we'll never know, because we didn't see the original river card. This is all part and parcel of the "random card concept" - as long as we are randomly selecting a card from the set of all unseen cards that could have come out, we are still being completely fair to both players. However, the mucked cards are simply not part of "the set of cards that could have come out". One of Chuck Ferry's really old books gives a nice nod to the randomization concept, and I don't see any reason why that wouldn't apply here.
Practically speaking, I think as long as you tell the players you are dealing a new river card from the undisturbed part of the stub at random, and tell them exactly what you are going to do before you do it, your decision is more likely to be respected. Tell them that the new card, just like the old card, has an equal chance of being helpful or not helpful to both players. Some possibilities would be:
1. Treat the previous cards effectively as boxed cards, and just deal the next card off the top without burning.
2. Burn the next card, and deal the subsequent card as the new river.
3. Shuffle the remaining stub (not the muck), and deal a new river without burning.
4. Shuffle the remaining stub (not the muck), burn, and deal the subsequent card as the new river.
I think all 4 of these options are consistent with the random card concept and are equally fair (or unfair) to both players, and I would not criticize any TD who chose one of these four options. Personally, with all the commotion and the delay, I would have leaned toward #3 or #4 in this case just because it may give the impression of "more" randomness in the dealing of the new river card, and all the players can view the shuffle. This may avoid accusations such as "oh the dealer just dumped the other cards into the muck because he knows what the new top card is going to be" or "how do we know the dealer didn't slip a new top card onto the top of the deck while all this delay was going on"? Unlikely scenarios of course, but better to be safe than sorry.