I think the biggest problem with dragging Leia into this part of the conversation is that it makes it seem like everyone, from the Skywalkers, Palpatine’s, and Rey herself, share the same blood. Everyone being related to each other by blood is like the last thing I want to suggest as it’s even more egregious than the original TROS. Not to mention the oncoming incest that would entail. That’s why I think we desperately need to maintain that the Force made them at this point, again. This is specifically why I worded Kylo’s dialogue earlier as “Palpatine influenced the Force itself to create life” and “You… are a rightful Palpatine”. The Force made the life, Palpatine is claiming what it made as his own because it wouldn’t have existed in the first place without him. This is an undeniable fact, even in canon. And who knows, perhaps Palpatine did boost them with an extra dose of “you need to fulfill your purpose with all your anger” because that’s what he always wanted them to do - kill him in anger. This is what’s hinted at in the scene between Rey and Finn on the Falcon, now. Rey says that she needs to destroy him in a trance-like state with Sith whispers in the background. So let’s summarize:

Kylo reveals to Rey that both of them only exist because Palpatine wanted them to. They are his heirs, destined for power, and must either fight each other for the throne, or put their differences aside and make their own dark order.
Luke says they aren’t Palpatine’s, they are the Force’s. They were always meant to end him. And apparently, end him through any means necessary since Vader took such a long time to do it. This is why he talks about some things being stronger than blood - not all offspring of the Force have the same willpower.
Rey confronts Palpatine armed with this knowledge. Palpatine explains that because of the hatred and anger in her for him (possibly planted there by himself), her killing him is what he always wanted for “his” creations. Even if she was never a Palpatine by blood, she nonetheless “will be Empress”. He’s basically stating that they’re destined to be family.
Rey’s willpower triumphs in the face of impossible darkness. Palpatine finally admits the truth: “You’re nothing! A scavenger girl is no match for the power in me” (this implies she DOESN’T have his power). She later claims herself a Skywalker as she is the offspring of the Force, just like them, and never belonged to her loser parents or Palpatine.