I need to understand what you are saying.
For instance, are you claiming that here:
residents on the west coast of South America and those on the East Coast of Australia would not be able to see the Southern Cross at the exact same time?
They may well be able to both see it, but their overall skies (not just one star in their sky) will be different.
Place a sphere, representing globe Earth, in a room. Call the four walls N, S, E and W. The ceiling is U for Up, the floor is D for down
Imagine observers at the equator, one nearest the N wall, one nearest the S. Both can look up to see U. However, if N is looking up, he will have E to his right and W to his left. If S also looks up, he will have E to his left, and W to his right.
Different skies for different observers, even though both can see the pole star (which could be at U or D)