oh really...dammit i wanna know D:i don't think it was ever mentioned....
I think you are about right since Nagato was also descended from that clan and Minato was already grown by the 3rd Ninja war meaning Kushinada was also grown. I assume the first Ninja war was during the period of the first and second Hokage which means Uzishiogakure was still in existence by then as the first's wife was from this clan.Uzushiogakure (Uzushiogakure no Sato; Literally meaning "Village Hidden among the Whirling Tides") was the shinobi village of the Land of Whirlpools. Its ninja were renowned for their fūinjutsu to the point that it led to its destruction in war.
However there was never any mention of the name of the war that destroyed the village but my guess would say it would of been the 2nd Great Ninja War