Would this mean that there would be points at the end of Antarctica where there is no South Pole gravitation or are the bar magnets basically a bunch of pie slices (meaning the southern end gets wider and wider as it reaches the end)?
The diagram is idealized and intended to convey a property of the distribution of the magnetic material within the Earth. A bar magnet is merely a collection of aligned polarized molecules all stuck together in a rectangle. If you have a collection of aligned polarized molecules
not stuck together in a rectangle, you still get a conglomerate magnetic field. The analytic way to view the diagram is to think of each bar magnet as having angular width dθ and allowing dθ to go to zero, while proportionally increasing the number of bar magnets.
To answer your question, however, it suffices to say that the magnetic field induced by a bar magnet does not cease to exist at/past the ends of the bar, so yes, there would still be a magnetic field.
The South end gets wider and wider, which is not what we observe, so Eramus theory is obviously false
What? Are you saying you dug up a north-south aligned, continent-sized bar magnet, followed it all the way to Antarctica, and observed that it did not indeed get wider as you went farther south?