It’s not that Rey cared about her grandparents, it’s that the reason her Palpatine heritage is such a big deal to her, personally, is because she is related to the Sith Lord who murdered trillions; on top of that, it’s part of the reason she exiles herself onto Ahch-To until Luke pointed out that she is still valuable and can still do the right thing regardless of her heritage.
But you yourself said that she had already overcome her need for her parents by TLJ? See, it’s right here:
Because of this, she finally overcomes her “need” for her parents, learning to move on from and stop caring for her parents entirely — hence why she seems happy and unaffected during the Battle of Crait; now, with no other option, with Luke having rejected her, with Kylo Ren being the bad guy, with Han Solo now dead, with her parents having thrown her away like garbage, she decides to attach herself to the Resistance and rely on them for validation.
…the reason she seems happy during Crait is because it means she has finally stopped caring about her parents themselves entirely.
If we’ve already come to this point, why is she all the sudden starting to care about her family again?