Gravity Paradox

    Occasionally I enjoy contemplating a universe with only two bodies -- originally to address the questions of universal coordinates and space-filling aether, later to consider implications of LIGO waves, recently to see whether I would predict perihelion precession -- this time to see whether I can account for stable orbits.  As will be seen, I can't.
    Owing to their gravity fields, the two bodies would either crash together or orbit each other.
   Consider the simple case where the two bodies are of equal mass and they orbit each other on opposite sides of a circle as depicted below.

Fig 1.  Model of two equal bodies orbiting each other.  The sunbursts at top and bottom of the orbit circle show where the bodies are now.  The small circles at upper-left and lower-right of the orbit circle show where the bodies were earlier when forming the bits of gravity field reaching the bodies now.  Arrows toward the small circles and away from starbursts show the direction of the same bits of gravitational field then and now, respectively.  Dash lines show the paths of those bits between formation and action.  Vertical and horizontal arrows from starbursts show centripetal and tangential components of acceleration, respectively.

    As I began to construct this picture, I realized that the gravitational field experienced by each body now is that diverging from the other body earlier when it was behind where it is now.  As depicted, the gravitational acceleration is not purely centripetal as needed for orbit stability.  It is mostly centripetal, but there is (according to this picture) a small tangential component in the direction of the body's orbital motion -- exaggerated here for illustration.
    Naively viewed, the picture implies the energetically impossible continual acceleration of each body in its orbital direction --  at no cost to the source of the acceleration.  This paradox leads me to suspect that a gravitational field might coast along with the body from which it diverged, thereby accelerating the other body centripetally purely.  But would that field coast as if following the circle or the tangent?  By what mechanism?  Could it be that gravity's acceleratory force propagates in a medium (aether) that is pushed and/or pulled along by massive bodies?  Or could it be that the graviton doesn't leave the source until it impacts the target, since time cannot progress in a light-speed agent.
    Something amazing about the mechanism of gravitational-field propagation might derive from these considerations.
    My theory of gravity doesn't involve gravitons, , , yet:
 https://ethicsblackhole.blogspot.com/2020/03/mechanistic-theory-of-gravity.html