I think Hagoromo realised that he was going to have to trust both Naruto and Sasuke one way or another. Yeah, he knew Sasuke was planning to kill Naruto, but he may have believed that he would change his mind, and if he hadn't given him power and everything to Naruto (who I'm sure could've taken Kaguya by himself in that case), then the vicious cycle of his children's transmigrants waging war would simply continue, and like Naruto inferred when he met him, this, at the very least, bothers Hagoromo as he essentially witnessed his children killing one another time and time again.
Sasuke would still try to kill Naruto, with or without Hagoromo's power, and if he failed and died, then the next transmigrant would give it a go, and the next, as would the next one after that, and so on.
Hagoromo has faith that Naruto will not only show Sasuke the right way, but that he will also succeed in breaking the vicious cycle started by Ashura and Indra. If he could make a friend and ally out of Kurama, the very embodiment of hatred, then surely he could do the same with Sasuke?