I don't think A'Xiu asked President Tao in either life to sign Yanyan over. What I think is more likely is that Tao just knew about Yanyan and took interest in her, because of her connection with A'Xiu. Then she either signed her over because of her talent, or to keep a closer eye on her, or - and this is my preferred theory - because she discovered Ningwei's fanfiction about the two of them.
I think it's more interesting if she did sign Youyan on due to Jing Xiu tbh. As we see in this chapter, Youyan's pride and righteous resistance to needing to suck up to other people was one of the reasons that she distanced herself from Jing Xiu in the first place. Obviously relying on connections, selling your body, etc these are all horrible things that entertainers shouldn't have to do. I think for a story as grounded as this, it's better to face these issues honestly, and not just do the cop-out "no you were special and different".
This is a moment for Ji Youyan to wrestle with these really ugly feelings, and try to come to a better conclusion than she did last time, which was drinking herself to death alone in an apartment. Like everything else in this story, their relationship and break up never feels contrived precisely because they had real flaws and problems with each other. I have no idea what the best response to this situation would be, but if she's committed to saving her relationship with Jing Xiu she has to face this stuff head on.