Aaaaaand there we have it...Yamao4nakama in the mud (even more so).
Momonosuke: W-Who are you...? You said my father's name before...!
Yamato: My other name is Yamato!!! I can die for you!!
As expected, Yamao's main relationship is very likely going to be with Momo. Not the straw hats. Yamao is declaring that she would die for Momo. That kind of devotion wouldn't make sense if at the end of the arc she left to join the straw hats, whilst Momo was still a young and defenceless lord with the duty of rebuilding Wano.
Yamao and Momo have a ton in common and I expect we'll continue to see them bond since after the arc Yamao would still have a role to play on Wano. Either as a co-ruler, a daimyo or whatever. They were the same age but because of time travel Yamao's 20 years older than him now. She's a parallel to Momo.
Yamato: But sorry!! I couldn't do anything but to see you as my father was about to let you fall.
Sasaki: !!?
Yamato: I didn't have strenght that day!!
They were both the same. Powerless to protect those they cared about. But now, Yamao has the strength to do so, whilst Momo still doesn't. I can easily see her playing a role at Momo's side after all of this. Their bond is only going to get closer and closer as the arc goes on and they discover their similarities. Resulting in Momo formally introducing Yamao to Wano's people and bestowing upon her a new role working with him.