[] There was no need for it. The character for Ling meant 'Zero' and that is what she had started as. If there were any lineage she would claim, it was her teachers, claimed through her own effort.
Honestly after thinking this for awhile this has for myself a few too many similarities with the direct subordinates of Shenhua not caring about the Past, and focusing on the Future for me to pick. Furthermore we can look to the CRX interlude about Xia Lin being rootless as for how such things can be important to others, and I'd expect there to be plenty of clans who place important in their history, which is a perspective LQ is liable to have issue relating to with this option.
Because whatever end goal Ling Qi decides she wishes to achieve in life it is the nature* of cultivation to force the cultivator to reconcile the reason they cultivate with what those who came before left behind, and those who exist in the present.
*As CRX noted to LQ the ripples a cultivator causes through simply existing scale with their cultivation level.