The first essay is by far the best introduction to TDT-like reasoning that I've ever read. In fact this paragraph sums up the whole informal part of the idea:

This solution depends in no way on telepathy or bizarre forms of causality. It’s just that the statement I’ll choose C and then everyone will, though entirely correct, is somewhat misleadingly phrased. It involves the word choice, which is incompatible with the compelling quality of logic. Schoolchildren do not choose what 507 divided by 13 is; they figure it out. Analogously, my letter really did not allow choice; it demanded reasoning. Thus, a better way to phrase the voodoo statement would be this: If reasoning guides me to say C, then, as I am no different from anyone else as far as rational thinking is concerned, it will guide everyone to say C.

Hofstadter's comparison of "choice" and "reasoning" is getting at the idea that people have decision routines rooted in physics, which can themselves be reasoned about, including reasoning that they are similar to one's own. I think this is really the core insight of the TDT idea.

And then the one-sentence:

Likewise, the argument "Whatever I do, so will everyone else do" is simply a statement of faith that reasoning is universal, at least among rational thinkers, not an endorsement of any mystical kind of causality.