“What if Sparrow decided to go ahead put herself on Patrick’s Contract” would be a heck of a canon-divergence side story.
(There’s a summarized version elsewhere in the archive. And in the Master’s Edition.)
But she is self-aware about her…uh, let’s not say “flaws,” because they could just as easily be advantages in the right situation…her personality traits that aren’t really suited to having this kind of authority over a Being.
Sparrow: Oh, good. “Patrick” it is.
Okay, Pat, let’s make a deal. If you had a “master”, they could order you not to make trouble. And to take off if trouble came looking for you.
I know that bit’s true. We went over it in public policy class. And if you can’t trust college textbooks, what can you trust?
That would take your threat level back down from “strange dude with magic powers” to “well-trained, if chatty, dog.” Then you could stay.
Patrick: Y-you want to become my–?
Sparrow: Noooo. Not me. Bianca’s much nicer than I am. Whatever agreement you make, you make it with her.
… she got one being later anyway.