You mentioned 'higher accuracy of Ancestry's test with regard to distant matches,' and I wonder about the other aspect (which you mentioned-sharing multiple independent sets of ancestors). I also wonder about another relative aspect (pun sort of intended): what happens when you share mutliple sets of ancestors? Is there a tendency to match up with those people more?
I've found a higher frequency of matches to third or fourth cousins who have multiple common ancestors, i.e, we have many common (independent) ancestors. AND I've found a higher frequency of matches to cousins who have the common ancestors several times over.
As a matter of fact, it is only the branches where I have mutliple common ancestors for myself, that I have located cousins through atDNA.