Hades is a carefree amnesiac Minecraft player. He has memories of his previous life, but those are all jumbled it would make no sense to actually think about it. He is an in the moment sort of guy who spontaneously gets ideas which are mostly inspired by Minecraft. He only really cares about those that are close to him. So he will not go after Kronos to save his siblings as he sees them as unimportant. He does not care about Rhea either as he only knew her for a few minutes. Hestia is realistically his only family as he sees it. His inaction will have consequences with his family in the future.
The Underworld is its own realm. He has an affinity to the underworld as his time in the void which was an underworld and most of his energy comes from and acknowledges him as its ruler.
This will be in the Riordanverse.
The basis of the Underworld being its own realm is the fact the Hades in the books was never truly acknowledged by the realm itself that he was the king and Tartarus was the closest it had to the king. Tartarus knowingly kept it secret that the Underworld was a different dimension allowing him full control over it. Tartarus is right now just seen as a prison not a place to torture the wicked so he is not condescending and all that. Hades now has complete privacy to operate in.
Hades was the ruler of the dead and that was all he ever was. He may rule the Underworld, but he never actually did anything else with it. He just saw it as an unfair burden in the books.
Hestia for his time in the Underworld will be his rock and tether to the rest of the World. Like I said he is carefree and is basically a newborn with facts and information that is not useful to him. She will take a little sister role in this story and hopes to be a family again with Rhea and Kronos. She does not know about her other siblings.
My character Hades will have many ideas about the underworld that will mostly be inspired by Minecraft. His soul is from outside the universe and he will not need to abide by the Ancient Laws. That is how he was able to enter the Underworld through a portal because he doesn't need to ask for permission unlike what Nyx assumed that he was already made king of the Underworld.
Hades is not an idiot he is just carefree and has no responsibilities. He is like a primordial without the responsibilities of being one yet. For now, he will make the Underworld into his image.