Jeff's thesis step by step resolved the open question left by the sos lower bound for planted clique (poetically, part of my PhD thesis): bounding the spectra of low degree matrix functions of ultra sparse random graphs (his theses shows applications far beyond sos lower bounds)

I taught the version you wrote. To be honest, I taught whatever appeared in these great notes by Sepehr and assumed that it was the algorithm in the paper. I'd trust you on the history of this algorithm, I'm only masquerading as a property tester. ;-)
sepehr.assadi.info/courses/cs51...

FWIW, my goal in bringing up Avi's excellent book and the last chapter that I quoted in particular was to clarify that "useful" as used in Shriram's post could/should be interpreted more broadly.

Parallel pancakes, or, a stack of pancakes by another name?

sounds like a setting that must have a phase transition... :-)