I'll first note that when thinking about the possible trajectories that reach a given spacetime point (e.g. earth in 2023), there is a trajectory that maximizes the elapsed time since the beginning of the universe.

Does that trajectory correspond to the CMB frame? If the universe is homogeneous, then yes. In the presence of peculiar gravitational fields, like those of our galaxy, that doesn't generally remain true, although I think you could pick a "time threading" such that it does. That's another gauge freedom I didn't mention. Along with the freedom to pick the "different positions at the same time" surfaces in spacetime, there's also freedom to pick the "same position at different time" lines.