Why did Minato's nine tails help in the fight?
Dark Kurama (the half sealed within Minato) has a good reason to help them. His very existence is under threat by Uchiha Obito (and later Uchiha Madara) Eternal Tsukuyomi. If Obito's team win, he will be sealed into the Gedo Statue and returned to being a part of the Juubi. Regardless of whether the tug-of-war between Dark Kurama and Minato happened or not, Dark Kurama would still help Naruto's side so that at least he can still be him.

With the tug-of-war taking place, he is even more willing to help since he's been tamed by Minato.