I'm afraid that Dalai Llama doesn't understand what having a good heart is. Pagan religions may teach man's fallen state while others deny original sin. Buddhism teaches man is in a fallen state but teaches no grace. It teaches self-achievement towards enlightenment. Buddhism is all about works salvation. I don't think Dalai Llama understands the reality of the wickedness of the human heart. Jeremiah 17:9 says that the heart is wicked above all things. If the heart is so wicked and man is so wicked, how can man's own effort clean that heart? The answer is you can't clean your own wicked heart with the effort of your wicked flesh.
The solution is only found in God's grace. Ezekiel 11:19 and 44:7 says that God will replace the wicked hearts. What's so sad is that people try to clean their sinful hearts with everything especially through religion. Buddhism may offer every mantra to remedy the heart, claiming that doing this and that would empty all the hatred and bitterness from the heart. On the contrary, it's relying on one's sinful flesh to get rid of one's captivity from hatred and bitterness. That's why God offers that only He can perform that heart transplant. That heart that's hardened by sin can be replaced by a new heart that will desire God.
Unfortunately, Dalai Llama doesn't really understand true religion. James 1:27 says that true religion is doing what's right and to be unspotted from the world. He's right to talk about the pure heart but does he even know how wicked the heart of the unsaved is? He doesn't. It's best to pray for his salvation right now before it's too late.