Relativity says that as motion becomes very much slower than the speed of light, behavior becomes very similar to Newton’s laws. Everyday materials (and planetary systems) and energies give rise to motions very very much slower than the speed of light, so it tends to be very very difficult to tell the difference. For a mechanical experimental design that can accurately described in a nontechnical blog post and that you could reasonably imagine building for yourself (e.g., a Foucault-style pendulum), the relativistic predictions are very likely to be indistinguishable from Newton’s predictions.
(This is very much like the “Bohr correspondence principle” in QM, but AFAIK this relativistic correspondence principle doesn’t have a special name. It’s just obvious from Einstein’s equations, and those equations have been known for as long as ordinary scientists have been thinking about (speed-of-light, as opposed to Galilean) relativity.)
Examples of “see, relativity isn’t purely academic” tend to involve motion near the speed of light (e.g., in particle accelerators, cosmic rays, or inner-sphere electrons in heavy atoms), superextreme conditions plus sensitive instruments (e.g., timing neutron stars or black holes in close orbit around each other), or extreme conditions plus supersensitive instruments (e.g., timing GPS satellites, or measuring subtle splittings in atomic spectroscopy).
I live in Plano (i.e., for y’all far away, a bit north of Dallas). I might be interested in participating in a meatspace study group arrangement of some sort. I’ve never done something like this outside of university classes, dunno how it’d work out, except to guess that it probably depends strongly on individual personalities and schedules and such.
I’ve studied parts of the Jaynes book in the past. Recently I’ve been studying more specialized machine learning techniques, like support vector machines, but it seems clear that more time spent studying the more general and fundamental stuff would be time well spent in understanding specialized techniques, and the Jaynes book looks like a good candidate for such study.