In other words, if we end up admitting that there's gravity between the Earth and the moon
I was thinking about this the other day and was going to propose a model for the sun and moon's orbit above the earth. As we seem to conclude, the earth may not have a gravitational field, itself, but that doesn't mean that other objects don't.
So, I propose a model by which the sun and moon orbit
eachother, around a common barycenter. This would explain their movement above the earth, as well as the trigger for the tides (or the cause of the oscillation).
Now, concidering the sun/moon mass as a single system about the barycenter, I would also propose that our immediate celestial neighborhood acts as a giant damped, spring-mass system of the form of a 2nd order ODE. The damping arises from the S/M's gravitaitonal attraction with the earth (or portions thereof), the spring force from the 'Dark Energy' (shmavity, UA, whatever) and the mass obviously from the system.
This set up could explain why the barycenter is at a constant altitude as the harmonics of the system would have died out long ago, but the actual orbits of the S/M are still sinusoidal about the center of mass.
The details have yet to be worked out, at present, this is just a thought experiment.