The five High Kingdoms of Man were established in Third Age 1 under the auspices of the Compact. With their permission, the Fellowship of Seven imbued the founding members of the five families responsible for these kingdoms with a geas-bound gift.
The transfer of the gifts to descendants in the family lines is not linear. A gift will be spread between all direct descendants of the first forbear who swore oath to the Fellowship. It does not pass in equal measure, but gravitates towards offspring who already have a natural tendency towards that gift. If there are a large pool of offspring to share the gift, it will be more prominent in some, and less pronounced in others. When there is only one descendant, the gift will fall on that one and be expressed with virulently full force.
Therefore the degree to which a royal gift expresses is twofold: inherited potency (how many descendants there are who could "carry" the trait) added to personal choice and character: how is each individual inclined toward that gift in the first place. In Arithon's case, his character is in accord with the gift as he is the only one living bearing the lineage. In the case of Lysaer's father, given the man's extreme hatred, and the long-term erosion of his character in the course of a truly bitter feud, the gift of justice was at play, or Arithon would have died in a messy way, rather than banishment.
The royal gifts do not transfer to caithdeinen in case of failure of a lineage. However, the potency of gifts adjust immediately when one member of the royal family dies: "he gasped under the shift as the gifted endowment bound into the royal bloodline transformed him with a magnified force that upended every priority."