¿Sorry? I am not talking about the stars disappearing in he horizon when you travel south. I am saying that people in Chile and New Zealand can see the same stars, at the same time (yes, in Chile they see them over the west horizon, and in New Zealand they see them over the east horizon, but at the same time). So, you are telling me tha in Chile cannot see stars that mexicans see, but can see stars that are seen in New Zealand, which, according to your Flat Earth map, are much more distant! How can this be possible?
Also, according to your wiki explanations, stars move like the sun and moon do. In the north pole, we see the Polar Star fixed. Stars near it move, but only a bit. As we go far from the Polar Star, stars move in wider circles (and faster, as all stars move at the same angular velocity). Then, when in the south hemisphere, stars near south must move faster than stars near north... But this doesn't happen!! As we look towards south, the circles in which stars move decrease, so their movement seems slower. ¿How can this be possible, according to you? And if you try to tell me that not all stars move at the same angular velocity, try again, because Ptolomeo called it "sphere of fixed stars" for a good reason: stars always are fixed respect the others. And even not accepting this, if stars near the south pole were slower than those near the equator, they could not be seen by the entire south hemisphere in one night, like it happens.