Opposition to Galileo was scientific, not just religious

General Relativity does not have global frames of reference. Borrowing from Special Relativity, an inertial frame of reference is on in which the Poincaré isometry group applies at every point in the frame of reference, and that means that fundamental physical constants do not change with translations, rotations or Lorentz boosts. That is, as you move back and forth along the spacelike axes, or rotate clockwise or counterclockwise around them, or acquire boosts, or move futureward, some set of constants are invariant; in pure SR the constants are the spacetime intervals -- the distance from one event to another. In the Standard Model of Particle Physics, there are other invariants, such as the fundamental charge and the speed at which a massless particle moves.

Special Relativity is a theory with global inertial frames of reference. General Relativity has no global frames of reference at all, except in the (flat, Minkowski) spacetime of Special Relativity. Instead, GR has the concept of the LIF -- the _locally_ inertial frame -- which exists everywhere in the tightest possible definition of local (specifically, in the local section of the fibre bundle, or in the limit where spacetime intervals go to zero), but only exists in _some_ bigger regions of spacetime (those that are flat -- or at least effectively flat, in the Wilson sense of an effective field theory).

None of this has to do with sets of coordinates; even in SR you can choose among a variety of coordinate systems (e.g., polar coordinates, Cartesian coordinates) and place the coordinate system's origin anywhere. Using different coordinates even in the spacetime of Special Relativity can help focus on aspects of a system under study, and indeed one can even use Rindler coordinates in Special Relativity to consider systems that include acceleration, or more often to treat a uniformly accelerating object as at rest at the coordinates' origin point.

In GR covariance is introduced to express all physical laws in arbitrary coordinates. Arbitrariness here certainly includes systems in which spacelike _coordinate_ distances (or the period of an oscillator) vary with position, or even in which the speed of light _against a set of coordinates_ varies with direction.

The Einstein Field Equations are written in a form that is independent of the choice of coordinates; they stay in that form even under a change of coordinates -- this is the generally covariant form. Newtonian physics, for example, can become much more complicated when you make a change of coordinates -- in this discussion, in dealing with planetary orbits under a change from a sun-centric polar coordinate system, or even a sun-centric Cartesian coordinate system to an Earth-centric system. In General Relativity the same Einstein Field Equation (EFE) form is used.

The calculational price for covariance is high -- the EFEs are ten non-linear partial differential equations. Even today that's heavy calculating. So anywhere one could get a decently correct answer using Special Relativity or Newtonian mechanics, especially where one can make a good choice of coordinates, you'd choose to do that instead. Where bodies move slowly against the chosen coordinates compared to how light moves against the same system of coordinates, and where the distances between massive bodies is large, the errors will be small to negligible.

So, to recap, "Isn't the choice between geocentrism and heliocentrism purely one of convenience, and we can simply pick and choose whichever [coordinate system] is most convenient for calculation purposes?" Yes, essentially, but one has to be wary about artifacts of the coordinate system that vary under a change of coordinates. A classic example involves latitude-longitude coordinates on Earth: there is a coordinate singularity at each pole, but the singularity goes away upon a change of coordinates (e.g. to local East-North-Up coordinates, or ECEF coordinates, or Cartesian coordinates with any number of possible choices of origin). Likewise, epicycles in planetary orbits change or vanish under different systems of coordinates, and so are unphysical.

I changed "reference frame" to "[coordinate system]" there to capture what an observer is doing: laying out an observer-centric set of coordinates. No such set of coordinates is the correct one, but any observer can choose manifestly useless sets of coordinates or draw incorrect conclusions from the motions of objects against one particular set of coordinates (not just what system of coordinates, but also where the origin is).