Straight Outta Lynwood was a great album. One of the CDs that I took out of my case the most often as a struggling nerdling who was still a year or two away from having scrounged up enough spare cash for a secondhand iPod.
I agree with almost all of what you have stated, save for a minor nitpick: I frankly don't think most functional adults think about the Fields Medal, similar annual prizes, or the qualities of the innovations of their candidate pools. I also think that that's totally okay. I think among a certain learned cohort of adults it's okay to hope that, and I think it's okay to imagine an idealized world where having an opinion on this sort of matter is a baseline, but I don't think it's realistic or fair to imply that (what I believe handwavily to be a majority of) adults are nonfunctional for not sharing this understanding.
The cyclic group generated by e.g. RU has order 105 (so 210 total turns or 105 of each side, alternated). If you have some math know-how, check out [1]. If you don't, take my word for it: when I was a teenager playing around with cubes, I once had a similar experience trying to do the same thing you did - when I went relatively quickly it never returned to the solved state, but when I was very deliberate about each turn, I got the 105 result (not by counting back then, but by rough time estimate given the figure I just looked up). Both you and I probably accidentally threw in one or more double-turns (like a U2) in there, or undercounted and gave up well before the cycle had completed (I, too, had thought I'd made "hundreds" of moves).
I think it's possible that that may be an additional benefit (for Google), but to me it seems overwhelmingly more likely that the main explanation here is Conway's Law.
You've identified a potential strategy by which a computer can play like a 1300-rated player, but not one where it will "play like a 1300-rated human". Patzers can still find and make moves in your set of N (if only by blind chance).
I want to note for the HN crowd that the book is in the "just technical enough to inform yet not scare off the layman, but not technical enough for the practitioner" nonfiction subgenre. Critically, there are a number of finer details that DFW gets wrong; if you're mathematically inclined and intend to read this, I suggest pairing it with a printed copy of Prabhakar Ragde's errata document hosted by the DFW fansite The Howling Fantods ([1]).