I've been thinking about this since watching the movie. Just listen, okay?
Tagah was closely connected to Sonam's lost avatar. So closely that she couldn't kill him, even though he clearly had the ability and the will to exact bloody justice for murdered airbenders.
And who does he see after waking up thousands of years later? He'd literally feel déjà vu. Just look at Aang and Sonam. It's clearly not just about them being avatars and airbenders.
It's also about their desire to change the world for the better, despite the cruelty that reigns around them. And despite the decisions the world dictates to them. At the same time, Aang does what she once couldn't: he gives up the powerful staff, he does what she once wanted to do.
Avatar Sonam and Avatar Aang are connected, and this is what Tagah sees. He sees that her reincarnation, chooses what she would have done in another world and given the choice.
And this is precisely what angers Tagah, or at least motivates him. That Aang chooses to believe in the best, just like her, despite the wars and destruction.
Tagah was doomed; he was stuck with the strong, sweet Avatar again, Lol.