The time Black Zetsu was born was precisely the moment Kaguya Ōtsutsuki was sealed by the Sage of Six Paths. Yet a fellow who only appeared later on could recount the history from Kaguya's descent into the Ninja World down to her sealing with absolute clarity.
This could only mean one thing: when Kaguya created Black Zetsu, she had already crammed all her memories straight into his head.
The Sage of Six Paths had personally dealt with his younger brother and sealed his own mother—such a matter was obviously impossible to publicize. The only reasonable explanation for Black Zetsu knowing the entire course of events was that those memories originally came from Kaguya.
In other words, from the moment Naruto started bluffing him, Black Zetsu understood that Naruto was spouting nonsense.
It was just that Black Zetsu's own combat capability was not quite strong, and later, having spotted the Nine-Tails inside Naruto, he estimated the strength of both sides and figured there was no chance of victory in a frontal fight. Thus, that old schemer who had lived for over a thousand years simply went along with Naruto's words and acted out the part.
If he wanted to call him father, then he would call him father; acknowledging an ancestor was not such a big deal either. As long as he could finally release his mother, Black Zetsu cared not at all about such a thing as face.
By the time he found that Naruto's speed in collecting the Tailed Beasts far exceeded that waste Obito, Black Zetsu simply stopped acting altogether and directly began cooperating with all his might. As long as Naruto became the Ten-Tails Jinchuriki, he would be able to wait for the chance to rescue Kaguya.
As for the true control method of the God Tree,