What a wonderful theory.
This of course could be true but then again she might also have wanted Zuko back along with having an advantage with him around.
Azula is so complex and tricky that its hard for even the fandom to speculate her thoughts and actions.
I agree with Gligo about the mother part and also i would like to point out that, that was not only the part where she was sort of kind towards Zuko but you did make a pretty good point.
This is what I personally think:
Azula grew not actually personally hand on hand loved by anyone other then her father and mother and honestly her father’s love was toxic. And she THOUGHT Ursa did not love her.
Would be a good reason why she would want her brother back however during when all this happened, she also had Ty Lee and Mai. She had almost the entire fire nation love and praise her (tho thats not direct love) so it does not seem as if she actually would have needed (she would but Azula probably MIGHT think she wouldn’t NEED him) Zuko.
I personally think that she thought Zuko redeemed himself and so she decided to be closer to Zuko seeing as now they were ALMOST equal and not the childish and foolish small children they once were. But she also knew/had a hunch that Zuko would be useful to her in the future cause Zuko is easy to manipulate but Zuko in the fire nation, allowed the two siblings to somewhat start to bond. But thats all jist what i think.