naively, could any percieved symmetry in these images be the effect of constant iteration over some input, where the self-similarity emerges as a kind of artifact of the lens, and not of the underlying numbers? Like how everything viewed through a kaleidoscope looks kaleidoscopic, but this doesn't reveal hidden symmetries in the object in the lens.

I'm suggesting it because even when you use other iterative tools (method of differences, phase space analysis[1], etc) you can percieve symmetry in the images even over random'ish data that mainly indicates limited or periodic inputs, but the output of these images don't provide any net new information about number theoretic relationships. It does suggest fractals may be a kind of lens artifact and not an actual property of nature though.

[1] e.g. https://lcamtuf.coredump.cx/newtcp/