Here’s something to chew on, from Weinberg’s Gravitation and cosmology:
Actually, I think his example could be better, since the Earth is large enough that tides are noticable, but on a small asteroid, say, orbiting the Sun, the equations of physics are the same as in any inertial coordinate system.
Ok I can buy that because in free fall stuff follows a geodesic. But are you saying that Mach’s connection between the bucket and the fixed stars is gravitational? Or is it some other undefined connection?