See, in my opinion this is were you make an assumption. You are believing the doctrine of Total Depravity that says the unregenerate cannot have a good and honest heart. But the scriptures show the unregenerate can do good.
Gen 4:6 And the LORD said unto Cain, Why art thou wroth? and why is thy countenance fallen?
7 If thou doest well, shalt thou not be accepted? and if thou doest not well, sin lieth at the door. And unto thee shall be his desire, and thou shalt rule over him.
God himself said Cain could do well and he would be accepted. And notice also that God said sin would desire Cain, but that Cain would rule over sin.
So, man is not totally enslaved by sin. Yes, we all have a sin nature that pulls us toward sin, but we also have the ability to resist and rule over sin. We can't claim like the old Flip Wilson routine "the Devil made me do it!". That is an excuse for a man not to take responsibility for his own actions. No, we can obey God if we chose to do so, the scriptures show that a multitude of times.
Psa 34:12 What man is he that desireth life, and loveth many days, that he may see good?
13 Keep thy tongue from evil, and thy lips from speaking guile.
14 Depart from evil, and do good; seek peace, and pursue it.
15 The eyes of the LORD are upon the righteous, and his ears are open unto their cry.
16 The face of the LORD is against them that do evil, to cut off the remembrance of them from the earth.
It is not reasonable for God to command us to do good if we are not able. And the scriptures say we are able.
Deut 30:11 For this commandment which I command thee this day, it is not hidden from thee, neither is it far off.
12 It is not in heaven, that thou shouldest say, Who shall go up for us to heaven, and bring it unto us, that we may hear it, and do it?
13 Neither is it beyond the sea, that thou shouldest say, Who shall go over the sea for us, and bring it unto us, that we may hear it, and do it?
14 But the word is very nigh unto thee, in thy mouth, and in thy heart, that thou mayest do it.
The doctine of Total Depravity is true in that man of himself will not do good. But the word of God has been in the world from the very beginning. Men have always had some revelation of God and are therefore responsible. This is why in Romans 1 it says man is without excuse.
Rom 1:18 For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who hold the truth in unrighteousness;
19 Because that which may be known of God is manifest in them; for God hath shewed it unto them.
20 For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse:
Notice in Deut 30 that it does not say a man has to be regenerated to do God's commands, it says "hear it". This is how we are enabled to do good, by hearing and believing God's word. If you can hear it, you can do it. Notice in verse 14 Moses tells the people that God's word is not hidden from them, it is very nigh, in their mouths and in their hearts that they can do and obey it.
Yep. A non-saved person can do good if they hear the Word of God. Churches are filled with those who participate in good works although they aren't saved. Yes, they can even hear the Word of God and rejoice for time, then fall away. They can be "social Christians" and yet be dead in their sins. This is my point.
The scriptures never say unregenerate man cannot do good, they say the opposite. This is why we can justly be held accountable for our sins.