OK, now you're coming full circle. As long as you're going to allow for the determination of coordinates to include events for which there is no physical object at the time and place of the event, then you're simply imagining what a physical clock and ruler would determine the coordinates of the event to be, correct? And as long as you recognize that there is a 3D finite element grid with a previously synchronized clock at each finite element (as opposed to a small number of randomly located physical objects all at mutual rest) and you can interpolate anywhere else, then all you have to do is imagine that you have a very course grid with elements only at the extremities and you can interpolate to find the coordinates of any event anywhere you want. The final step is to simply imagine that even the elements at the extremities are also imaginary.
But I don't want to lose sight of the import of this thread which is your claim that only objects that are at mutual rest with a Frame of Reference belong to that Frame of Reference. I want to make sure you recognize that Einstein's definition of a Frame of Reference does not contain any physical objects, but an infinite number of imaginary objects (rigid rulers and synchronized clocks in the finite element grid) and you use these to specify the positions and motions of any number of physical objects. (Of course, in our thought problems, even these "physical objects" are imaginary, but we image that they are actually physical, whereas we image that the coordinate rulers and clocks are actually imaginary.)