Nutjob theory time: Aurora is the real daughter and Lapis, probably not her given name, was a commoner, her parents are in anti-noble faction, probably the Tenebrae Family's servants or employee, and tried to use her to harm the lil sis, but she couldn't betray her and the scheme failed so they abandoned her and fled themselves. Lil sis died protected her, the Tenebrae Family adopted Lapis for both politic, pity and love, so now she decide to devote herself to weed out the anti noble crowd and probably herself in the progress. Earlier chapters feel like she resent the commoner more than just maintain the status quo to me. The whole jewelry metaphor is not about Natalie but about her past self, being thrown away and then was given care and education by the noble so she found a purpose now, being the support pieces for the nobilities and the royal family in her own senses (the chess metaphor).
1st chapter had her being mocked and isolated by her peers so imo she tried to shoulder all this herself and not fuel class war between nobles and commoners. The anti noble probably will contact Natalie and expose Lapis' past and present atrocities and try the betrayal scheme again.