First, here some parallels between Oden and Yamato:
- Oden and Yamato both were/are “trapped” in WANO.
- Both were waiting for “Someone” to take them out to Sea.
- Both “yearned” to see the world outside.
- Both being the “Blood” of ‘Most powerful person’ in WANO.
- Both told the First Pirate they met They have been waiting for them.
- Both expressed They would like to Join!
“I am Kozuki Oden and You are Ace’s Brother!”
- The Key Person here is “ACE” – the guy who is son of “TWO” fathers! Both of his fathers adored Oden and treated him as equal!
- ODEN was taken away by Whitebeard and then Roger ~ on his Journey – of whose Journal is now “Bible” for Yamato.
So, initially the task should have fallen on Ace who was son of Whitebeard. Yamato carried the Will of Oden, she wanted to go away with Ace, but couldn’t because of cuffs. Later, She should have also learned that Ace is son of Roger too. Thus, she has been waiting for him, since Marineford War and demise of Ace.
Thus, her quote with Luffy, “Since I am Oden and you are Ace’s brother” can be easily interpreted as, “I carry the Will of Oden and You carry the Will of Ace/Whitebeard/Roger!”
That’s her logical conclusion that she has right by default to be with Straw Hats just because of Luffy is Ace’s brother. With Ace being dead, Luffy has to carry Ace’s Will.
As a daughter of a Yanko, surely Yamato is a formidable fighter. Can you imagine the strenght of the Straw Hats Crew with her addition?