Sorry to be a fly in the ointment Simon..
This essay is well-written, and your point is logically argued given your premises, but I can easily take some of them apart.
A lecture at the 10th Frontiers of Fundamental Physics conference (in '09 at UWA near Perth) by Mikhail Kovalyov stated that the Physics including the higher-order terms IS what's fundamental, so jerk is in no way a departure from fundamental Physics. He explained that non-linear terms are hard to deal with, and often lead to unsolvable equations, so people make limiting assumptions to have solvable equations, plug in numbers, and test their basic assumptions. It is unwise to equate basic with fundamental, though, especially in a topic like Physics. So I see your doing so as a big logical flaw.
Some of your statements about transient phenomena still make sense, and I love the discussion about fine-grained vs coarse-graining of phenomena, but it is a critical error to put an elevator jerk on a par with something like what Brian Greene talks about in Elegant Universe, where rips in spacetime are self-healing because catastrophic flop transitions are forbidden. As you say "You'll be reminded of the limits of your knowledge, but the universe will not catch fire" because transient phenomena are not an existential threat. However there are limits to your analogy.
I've been researching exceptions to your generalizations for more than 30 years, starting with a discovery involving the Mandelbrot Set, and a few phone conversations with Ben Mandelbrot back in the 80s. My essay this year focuses on the Octonions, and addresses some of the issue you raise. But the Mandelbrot Set is a wonderful example of a self-referential action in pure Math. Start with a (complex number) value, multiply it by itself, add back the starting value, then repeat. Each time the point of origin is referenced again, but that is the whole formula in a nutshell. But you end up with something that spawns endless copies of itself in miniature.
I have been filling in the gaps you describe as a student of many of the origin mysteries you describe, for some time now. I assure you some of the problems you talk about are solved, if you don't want to re-invent the wheel. But I'll sign off for now. It was a good read and a good try, Simon, but you simply get a few things wrong. I'll be happy to take as much time as needed to explain. Mind you; it gets kind of deep probing the ultimate origins, but that's why I attended almost all of the quantum gravity lectures at GR21.
All the Best,
Jonathan