Your logic breaks down at the assumption that "could have" refers to right now. "Could" refers to a possibility - in this case, she sings about the past in which the possibility of her becoming a princess was destroyed. Thus, as a possibility, and NOT the present, it remains in the future.
On top of that, you can't assume that all she wants is the title by a few lines of the song. It's like reading "Thank you for applying to our college" and assuming that that means you're accepted, but when in reality, the letter later says "We're sorry, but we regretfully say that we cannot admit you to our school."
Look at the song as a whole.
What you need to do is take into consideration what they mean by Princess and King. Don't take it literally. They're just metaphorical personas. What is the connotation of a princess? Girls who are called princesses are usually provided for, even spoiled, to the point where they demand a lot (however, that's not to say they act like brats, nor is that to say that all they want are material objects. Maybe the "princess" demands more time and attention from the "king"). What about a king? A man who rules over the nation and can provide. But maybe the king can't always provide for what the princess wants (maybe he doesn't have enough time to spend with her and give her the attention she wants).
What the king wants and what a princess demands can conflict. With enough conflict, the king can decide to disown the princess. In doing so, the family, home, and their unity are destroyed. Where does the song support this? "Could've had a castle, worn a ring." In this case, the castle represents the grand home/family they could have continued to have together, as also represented by the ring - unity in how the ring is continuous. The two even "fell apart"
The song also supports this theory in the fact that both artists sing "You really hurt me." The king was hurt because the princess, in demanding so much and conflicting with him, seemed to turn against him. The princess in this case could have been hurt by the fact that the king disowned her and left her with no place to go. All a metaphor for why the two lovers broke up - they couldn't meet each other's demands, so in conflict, they decided to break up, leaving each other hurt in the end.