Crazy Eddie wrote:
You wouldn't need to treat a curved space as if it was flat, you'd just need a mathematically consistent way to account for that curvature itself. Which is, like, stupefyingly difficult, but hardly impossible

The Minkowski metric is the
very definition of flat. It's simple: (1, 1, 1, 1)  apart from the minus sign, it's pretty much a straight map from number to space, no transformation, so if I feed x = 2, y = 3 and z = 3 into it it gives me back x = 2, y = 2 and z= 3.
"Minkowski space with curvature" isn't Minkowski space but something else. Once you introduce a set of coordinates with anything other than 1 and 1 as the metric, you're back in curved space. And special relativity comes out of pretty much all spaces, but only in locally flat coordinates and linearized gravity in that space thus making it a subcase of GR.
Crazy Eddie wrote:
Locallyflat spacetime is a mathematical conceit that doesn't actually exist in reality and so special relativity itself is just an approximation (like using the ideal gas model to calculate drag coefficients on a spacecraft; you can get away with it under some circumstances, but not all).

"Special relativity is just an approximation"  where's your experimental evidence for this? Hundreds of experiments have been done which have verified special relativity, taking place in locally flat spacetime. You've said dark matter theorists are getting ahead of themselves, but here you mention something which flies in the face of modern quantum field theory and astrophysics without anything to back that up.
"Dark matter", whatever it is, isn't just a mathematical invention done for shits and giggles. We have CMB observations from Planck, and observations of gravitational lensing which tells us that there's something going on there. It could be an undiscovered form of matter, it could be quantum gravity effects... but apart from the "dark matter" problem we know the rest of relativity is pretty much right.
This isn't "the establishment" talking, but rather experimental data. Special relativity has been verified time and again by RHIC, LEP, LHC and many other experiments. If an unusual gravitational effect had popped up at any of these that indicated something was wrong with basic special relativity, it would have been noticed and studied.
Nothing so far, and unless a spectacular deviation from special relativity at the colliders no reason to doubt relativity. Physicists don't just "accept" relativity  if, say, we discovered Lorentz violations at a collider we'd go and start reworking what we know about it.
Crazy Eddie wrote:
Really, I'm saying that Einstein's relativity is a paradigm that obfuscates the fact that Minkowski spacetime IS applicable to curved (or rather BUMPY) spacetime

This is a
speculation on your part, not a
fact. I'm a theoretical physicist: what I do is theory, and fact only if it pops up in an experiment. If I say something is a fact in a paper, then I've got to back it up. I can say many aspects of the Standard Model are a fact: they've been measured. Relativistic time dilation: a fact, it has been measured.
Academic language is full of disclaimers when applied to theoretical physics 
fact is a very loaded term to use, and here you're saying that a flat space metric is applicable to curved spaces which kind of flies in the face of geometry. The distance measures of Minkowski space when applied to curved space will not give correct results as they would fail to take into account the properties of that space  (1,1,1,1) would give wrong answers when mapping numbers to the manifold surrounding, say, a star or other spherically symmetric body.
If you can put forth a substantial body of experimental data to show that equations of motion in curved spaces can be found using the Minkowski metric, I'd accept that as a fact. I accept that there is something we call "dark matter" out there because of experimental evidence, although what the specifics of it are I would say are unknown.
When it comes to "the Minkowski metric can be applied to a curved space" I generally tend to trust my differential geometry books which define it as a flat space and only a flat space, rather than your statement that this isn't true.