This is getting closer to the solution, and this tutorial was helpful: Match features with symbols in a style
But the problem remains that I'd have to create and rename 500 marker styles and then define every single marker symbol (up to 500 points) within my style to correspond to the right PNG picture file. And then any time that I regenerate my attribute table with more earthquakes, the whole list could get out of sync, matching the wrong beachball symbol with the wrong earthquake.
To answer your question, the point of this symbology is to quickly see the style of each individual earthquake in an area. This is a commonly used technique in geology and geophysics because it instantly tells the viewer the faulting regime (i.e. strike-slip or thrust faulting), it hints at the orientation and relative magnitudes of the forces that caused the earthquake, it gives some indication of the nature of surface shaking that resulted from that earthquake, and it gives 2 possible orientations for the fault that slipped to create the earthquake. These are very powerful visual tools and it's relatively easy to map them in MATLAB, but apparently a lot more difficult to map in Arc. I'd much rather create a map in Arc than in MATLAB because Arc typically produces much cleaner maps in a less cumbersome way.