@Mathphile My own (possibly crackpot nonsense) ideas revolve around the idea that relativistic gravity can be better modeled as variation in density in spacetime, instead of curvature. (You can see that the ideas are analogous by considering that the surface area of curvature is proportional to the degree of curvature - that is, a bowl has more surface area than a circle)

If we treat that as the case, Lorentz Contraction can be modeled as the cause of motion, rather than a result, because gravity itself is getting contracted, which means spacetime is denser around a quickly-moving object. I believe there is a slight asymmetry to Lorentz Contraction, given that there is an asymmetry to the electromagnetic field in a Lorentz-contracted object (a photon is more likely to go forward than backward)

Which means that spacetime may be slightly more dense forward of a moving object than behind it, meaning in turn that motion through time results in motion forward, for the same reason objects fall into the Earth as a result of increased density in that direction (approximately, the side experiencing more density has "further to go").

Rindler Coordinates are also easier to understand in terms of density rather than curvature. (Rindler Coordinates are what Lorentz Contraction looks like under acceleration; namely, Lorentz Contraction begins to look exactly like gravity.)