To address motivation issues. Choosing to take the cai route comes from a few points in Ling Qi's character.
On the shallowest level, there is Ling Qi's desire for power, or more specifically, her desire to be able to control her own circumstances. There is no path that allows her to do that entirely... but Cai's is the one the that puts the fewest people in a position to order her around.
On the second level, there is her budding desire for a home. The Sect has been her first home, and could certainly grow to be a true one... but Cai is the fastest route to receiving a place that is hers and hers alone(mostly, don't sweat the legal details of vassalship, she's not thinking about those.)
On the third, it will Ling Qi deciding to engage with her low level desire for something higher than her own selfishness, she might not believe in Cai's ideals, but she is willing to give them a shot and see if they can survive reality.
Are plenty of those somewhat shortsighted and not entirely logical? Yes, because Ling Qi is still fifteen.