There is currently nothing the trash drops, aside from your durability, that warrants you killing them off. There's no benefit to it, they don't drop gil, and the items they can drop are split among party members in the round-robin fashion. If they gave us more incentive to kill off the trash then sure I might not find a way to bypass it, but as is, there's just no point in killing trash.

Not to mention letting Mythology stones drop in only one dungeon (I don't count the primals no one does those for fun or to farm stones) was just a poor design in general. It makes no sense to have great gear with only 1 currency that can only be obtained through 1 specific dungeon. Maybe when the new dungeons come out this will change, who knows.