Zagreus might also be able to permanently end monsters by eating their essence in percy jackson.
The thing about monsters is that they are mortal to celestial bronze yet resurrect. Chiron call them archetypes, this once again plays into the Two Worlds metaphor. Luke accuses the gods to repeat patterns and play the same old stories endlessly, he tells Percy he stole the Master Bold just to do something nobody had ever done before.
This also comes to the problem of immortality in both legend, PJ and VSons. Griffon says that "our faceless divinity could persist beyond death, it followed that our Heroes could do something similar. Perseverance, not through death as the divine might. But within it. A half remembrance. Lesser. Yet still profound."
In this case, where is Dyonisus who could not be destroyed by Hera? Double immortal like Su Wukon seven immortalities?
I decided to rule that VSons gods 'die' the same way Kronos did in PJ. It's more like they are broken machines, and like machines, they can be easily fixed in a way mortals can't. The only reason someone like Asclepius never 'fixed' them VS is because no man like Asclepius, capable of defying death, has emerged in the Iron Age.
In this case, Zagreus could kill monsters in a more long-lasting way. To the point that instead of perfectly reforming with old memories, a new monster is born inheriting the karma of the one he killed. In a sense, it is death as mortals do and it is just as frightening because it enforces great transformation. Some ambivalent and tragic monsters like Medusa would get a 'new lease', like how Percy lost the curse of Achilles, granting them a fresh start. Athena would recognize her but enforcing the same curse for a grudge is too much something Hera would do.