If we’ve already come to this point, why is she all the sudden starting to care about her family again?
Because Rey has no arc. Or at least, not a compelling one. Her ‘journey’ is some weird emotional tangent the movies sometimes decide to touch upon, and repeat in literally every single movie in some weird 360 tailspin.
TROS’ entire message is basically that self worth is found from within and you are who you make of yourself, as deducted by the OP. The issue with this is that her stealing the Skywalker name totally contradicts this and is nonsensical within the confines of the story, being done for nothing other than fan service.
If heritage doesn’t matter, why change your name? Why not keep the name Palpatine? Why have a last name at all? To honour someone you looked up to? Well you didn’t look up to Luke, you screamed at him a bunch for three days, beat him down and then stole his books. Then talked to him once after he died. You didn’t know him. Why not take Solo’s last name? That actually has narrative significance, however slim and also nonsensical considering you didn’t know the man for more than a couple days and his son was a genocidal lunatic who tortured you and killed everyone. But you did kiss him so I guess that’s all forgiven.
The trilogy is a mess of contradicting viewpoints and themes and Rey’s entire ‘journey’ takes her from being a family-less power-God who doesn’t need to earn anything on a desert planet, to a family-less power-God who doesn’t need to earn anything on a desert planet. Except now she’s got inner peace and acceptance, as clearly shown by assuming another family’s identity she doesn’t really know without their say in the matter.
Rey’s “journey” in comparison to Luke and Anakin is what I’d politely call baffling and undermining.