There are some parallels between Oden and Yamato;
1st- both are the sons of the supreme leader
2nd- both are called idiots by their father
3rd- character designs
4th- born rebel in nature
With these parallels in hand, I’m gonna add in some assumptions,
5th- since Yamato has hand cuffs yet he wasn’t on prison means he was not being treated as a prisoner but was being kept on check just like how Oden would have or might have been.
6th- maybe he wrecks havoc just like Oden so he needed to be kept on check
7th- he wants to explore the world as a free man and not be controlled by his father. He could have tried escaping before too which is why Kaido sent his best men on the job.
8th- met his first Wano outsider and the outsider was a Great Pirate.
To conclude, Oden always wanted to explore the world and same could be the case with Yamato. His traditional dresses may be the hint that he has never stepped outside Wano. He heard about a great pirate invading Wano and hatched a plan to escape Wano along with him. Oden’s first meet with WB and Yamato meeting Luffy also has a somewhat similarity. Oden asked WB to let him join his crew and Yamato too said somewhat similar with him waiting for Luffy all along.
Yamato for the next Nakama..!!??!!
P.s. when I said Nakama I meant like how Oden was to Roger.