To say that "you don't need God to do good" sounds to me like "the fruit can exist without the tree" or "there is a tree but it produces a different kind of fruit" which means "yes there is God but he is not good."
No one else is good but God.
Unbelievers do good things.
Yes that's true.
It is because humans are all created in God's image, unlike the other animals, we have moral accountability.
Now, doing good is not what will get us to where God wants us to be (Heaven). We all sinned and sin means death, separation from God.
We cannot buy our way back to God and so God reached out to us.
Man's attempt to please God is called religion.
God reaching out to man is called grace.
That is what exactly Jesus did when He became a man and suffered and died then resurrected for our sins. We can finally go back Home because our Lord paid the price for our sins.
Let this truth not depart from you my friends.
