Not only does it say that but I am afraid that is exactly what that means. Without being born again and having the Spirit in us there is nothing good we can do or think. Only the saved have the ability to do good.
I think your understanding is one of the problems in the church. The Pastors have not taught the people just how evil we are. There is nothing in us that is good apart from Christ. Even the good that we would do comes from evil intent so there is nothing good in us. Only that which is done for Christ will stand and is good as no lost person can do anything for Christ so they can do nothing good. Uless our motives are to glorify God even the best we do is a filthy rag, evil and no lost person can have a pure motive to glorify God.
And you err just like Aaron. Jesus himself said sinners do good to those who do good to them. This is not sin, it is not sin to do good to those who do good to you, you SHOULD do good to those who treat you well. Jesus simply said it is a greater good to be like God and do good to those who are evil toward you. Nevertheless, it is not sin to do good to those who love you and treat you well.
Cornelius was not saved, yet he did many good works that were recognized by God. God sent an angel to Cornelius to tell him to send for Peter whereby he and his house would hear words whereby they would be saved. So we know for a FACT that Cornelius was not saved at this point and unregenerate. Cornelius did not receive the Holy Spirit until AFTER Peter came and preached the gospel to him.
Cornelius proves your view to be error.
Acts 10:1 There was a certain man in Caesarea called Cornelius, a centurion of the band called the Italian band,
2
A devout man, and one that feared God with all his house, which gave much alms to the people, and prayed to God alway.
3 He saw in a vision evidently about the ninth hour of the day an angel of God coming in to him, and saying unto him, Cornelius.
4 And when he looked on him, he was afraid, and said, What is it, Lord? And he said unto him,
Thy prayers and thine alms are come up for a memorial before God.
5 And now send men to Joppa, and call for one Simon, whose surname is Peter:
Cornelius was not regenerate, but he was a very good man who feared God, prayed always, and gave to the poor. God heard his prayers and recognized his good works.
Acts 10:34 Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons:
35 But in every nation he that feareth him,
and worketh righteousness, is accepted with him.
Was Cornelius saved when Peter said this? Nope.
Acts 11:13 And he shewed us how he had seen an angel in his house, which stood and said unto him, Send men to Joppa, and call for Simon, whose surname is Peter;
14 Who shall tell thee words, whereby thou and all thy house
shall be saved.
Peter here confirms that Cornelius was not saved when the angel spoke to him.
Acts 11:15 And as I began to speak, the Holy Ghost fell on them, as on us at the beginning.
16 Then remembered I the word of the Lord, how that he said, John indeed baptized with water; but ye shall be baptized with the Holy Ghost.
17 Forasmuch then
as God gave them the like gift as he did unto us, who believed on the Lord Jesus Christ; what was I, that I could withstand God?
This is when Cornelius was regenerated, when he believed the gospel preached to him by Peter and received the Holy Ghost.
Cornelius absolutely refutes the false teaching that unregenerate man cannot do good.