Peter,
I've not spent that much time and effort finding problems with special relativity, but I seem to have more friends who find problems with SR than with anything else, and often, 'where there's smoke, there's fire'.
It looks like it will take a while to study your Jackson/Nixey pdf and I will get back after I've read it.
Because gravito-magnetism was initially believed to be too weak to be of any consequence, it has been essentially ignored for 150 years. In fact, the Phys Rev Lett paper cited above is possibly the first time that gravito-magnetism has 'officially' and unequivocally attributed the discrepancy with general relativity's predictions to the gravito-magnetic field (aka the 'C-field').
So my basic premise is that gravity G, gravito-magnetic C, electric E and electromagnetic B-fields exist, and I believe G and C are 'classical' ie, each field is a continuum. I am less sure whether E and B are continua or whether they are statistical effects of bosons. I'm thinking about it. Maxwell first wrote the GEM equations simply based on symmetry (Newton's ~ Coulomb's equation) and these were later derived as the 'weak field limit' of general relativity.
The key fact is that my calculations provide reason to believe that gravito-magnetism is 10**31 times stronger than was originally believed, and Martin Tajmar has experimentally found the same factor. If correct, this has very significant consequences for particle physics and cosmology.
On the surface Maxwell's EM equations and the analogous GEM look very similar. But there is a drastic essential difference. The EM fields interact strongly with charge, but are themselves uncharged, hence their self-interaction is linear and supports 'superposition' in the mathematico-physical sense of interference. But the GEM fields interact with mass and, through self-energy E=mc**2, thereby interact with themselves in a non-linear, ie, Yang-Mills manner, providing for physical phenomena that have been attributed to other fields (which physicists freely invent due to the nature of the Lagrangian technique).
If it turns out to be the case, as looks more likely every day, that the C-field is not only real, but has the strength I claim it has, and, as also looks more likely, neither the Higgs nor SUSY particles show up, then "somebody got a lot of 'splainin' to do."
I claim the C-field explains dozens of anomalies that GR, QED, and QCD cannot explain.
The results of this, if true, will be, as you say, that physics is much simpler than is currently believed to be the case. This, incredibly, will make a lot of people unhappy. Go figure.
Edwin Eugene Klingman