The way that the Epstein Family would test for the next family head was by testing one's bloodline, talent grading and elemental affinity all at once using a special crystal that the ancestor of the family had left behind. He also left behind all his research on runes, pill-making, arrays and weapons and was the one that gained the most from the destruction of Elizabeth's family.
With all of that research and knowledge that was left behind, the Epstein Family continued to develop as the most knowledgable family and was the family that Elizabeth hated the most because of it.
It wasn't a rule that only the descendants of the family head could use test themselves using the special crystal, but every time somebody tried, they would always be below that of the family head's descendants so it became an unwritten rule to do so.