The reason this did not behave as expected is because the receiver activeTileQueue (the NSMutableOrderedSet instance) already contained the member.
While I specifically sought the use of a set versus an array because I didn't want duplicates of any objects inserted, my assumption about how the insertion would work in this case was incorrect. I assumed that if the set already contained the member I was attempting to insert, that rather than be re-insereted, the member would be repositioned to the index passed in -insertObjectAtIndex. This was incorrect.
If the receiver already contains the member, it is not re-inserted or re-ordered. In short, nothing happens. To obtain the originally desired behavior, I would have to either remove the original member and re-insert it, or use another method on NSMutableOrdered set to reposition/exchange the index within the set, eg:
NSIndexSet *replacementSet = [NSIndexSet indexSetWithIndex:[[self activeTileQueue] indexOfObject:recentObject]];
[[self activeTileQueue] moveObjectsAtIndexes:replacementSet toIndex:0];