I do not agree with that Yamato is like the "princess of the arc". Traditionally the princess in one piece is usually damsel in distress who cannot really fend for themselves without Luffy or any strawhat being able to inspire them to be better leaders. Most princess that you just name were crybabies and that do not have self-confidence in themselves. Some of the princess in those respect arcs were well love and care for by their families. An individual who represents the princess trope that you just mention would be Momonosuke. Yamato unlike other princess does not consider the place she was raised to be her home.
Yamato has lot more common with Sanij than any other princess. Both had strict and abusive fathers that had high expectation for their child to be something greater for them. They were both prisoners under their father’s organizations, had explosives cuff on their wrist that prevent them from leaving their family, hate dads and hate their bloodlines that tie to them their fathers and let us not also forget they were both starve to death. Unlike Sanji Yamato did not have any other relatives who really showed her compassion. It was samurai, ace, and the random beast pirate who showed they care about her wellbeing. However, the thing with Ace though, is he did not have advance form of Haki to free her and could not keep his promise to liberate Wano. Sanji had one sibling that care about him, and he was free from his father grasp. When he met zeff he met someone with common goal like he did. Luffy defeating Kaido allows Yamato to live her life.
Finally, Yamato unlike most princess she was mostly eager to meet Luffy. She was at least able to by sometime until Luffy had arrive that both King, Queen and Jimbei acknowledge her importance within the arc. She is not at all interested babysitting Momonosuke and being one of his retainers. As well she does not want to take over her father’s organizations, she wants to be a service for Luffy and the rest of the strawhats. Princess in one piece they were not trying to join the strawhats and leaving the place they call home. They just wanted things go back normal or zero. The only thing that princess usually suggested to make promise that they meet again, would they consider them a friend of theirs or other additional promises .