We have Quakers, including Stanley and Hutchins, but did not get those pieces.
I tested myself and my brother at FTDNA. We have about 260 some in common matches and we each have about 460 not in common matches. We share the expected percentages of DNA for full siblings, the same mtdna, and the correct percentages with our father whom I also tested.
We each have the same relationship to those other over 900 people, but did not come up as matches. I never expected so many not in common matches.

I love math and statistics, and tested my son at 23and me, and his father at FTDNA. I also tested Y and autosomal a known 3rd cousin on my father's side. That cousin is a Y-match and a cousin match for 2nd cousin once removed, my father, but not a cousin match to either 2 of his known third cousins, myself brother. My husband also has a 3rd cousin and their match is just under the 7cM recommended by GEDmatch.

Toward a more distant connection, my husband and a known 3rd cousin did Y-testing as Hodges. They match each other, but after that match only the surname Crow. The cousin match turned up 2 Crow cousins with some of the Y-DNA matches, and then additional female lines with these Crow families. One of the Crow Y-DNA matches is not a "cousin" match, but his sister and 2 of his cousins are. These people could not be closer than 5th cousins, and looking at their lines we have not found a possible connection even that close.

What I have observed is DNA is very random. What passes can remain an intact segment from grandfather through a daughter to a grandson or become truncated. We know 1/2 is lost each generation. The chances that 2 cousins inherited the same part decreases drastically with each generation, even as recently as 3rd cousins. Triangulation is, like you wrote, exciting.

Joyce (Moore) Hodges