In special relativity every frame has its own unique time axis, represented in Minkowski diagrams by a fan-out of time vectors that grows infinitely dense as you approach the surface of the light cone for a given rest frame. Viewed from any other frame, these same time vectors shift about exactly the same fan-out topology, so no one frame perspective is any more "correct" than any other one.
In general relativity this does not quite seem to be the case. If nothing else, all those worldlines extend over billions of years and so presumably end up having a quite precise average time orientation for any given local region of space, even if you select a complicated overall topology. Conversely (and I assume equivalently), if space is viewed interpreted as manifold, the idea of an average "flatness" that is orthogonal to this long-term time axis appears to be assumed, since it would look quite odd to see the entire universe slanted relative to its own time axis, yes? So:
Am I understanding this correctly? Does GR applied over eons necessarily end up defining and distinguishing a unique and well-defined overall time axis?
If so, how is this unique time axis ascertained in experimental practice? Is it taken to be some sort of very large center-of-momentum average, as the billions-of-years bundle analogy would seem to suggest?
If a distinguished time arrow does exist in GR, does the existence of that axis imply any experimentally meaningful implications for its associated SR frame?