The children who came of age in a population where the worship of created things was firmly established know better than anyone that the worship of created things is a waste of time and effort.  At that moment they have an opportunity to turn back to the worship of the living God, but if their ἀσέβεια (ungodliness) is not overcome by the Spirit of God, the word of God, faith in the Lord Jesus, and being born from above they are more likely to take the next logical step.

I think the King James translation of the first part of Romans 1:28a can really help here: And even as they did not like (ἐδοκίμασαν, a form of δοκιμάζω) to retain (ἔχειν, a form of ἔχω) God in their knowledge (ἐπιγνώσει, a form of ἐπίγνωσις)…  Paul used the word ἐχέτω (another form of ἔχω) in the sense of having a wife or a husband in 1 Corinthians 7:2 (KJV): let every man have (ἐχέτω) his own wife, and let every woman have (ἐχέτω) her own husband.  Considering where all this is leading—you also died to the law through the body of Christ, so that you could be joined to another, to the one who was raised from the dead, to bear fruit to God¹—the double entendre of ἔχω intrigues me.  It is something like, even as they did not approve of having God in their knowledge.

Approve is both a possible and likely meaning for ἐδοκίμασαν (a form of δοκιμάζω) in Romans 1:28.  It comes from δόκιμος.  And the negation of δόκιμος, ἀδόκιμος (ἀδόκιμον), is the unapproved (NET, depraved; KJV, reprobate) mind the Lord gave those who did not approve of having Him in their minds.  All of that beautiful logical symmetry is lost in both translations in favor of emotionally charged language that fails to communicate that the mind He gave them over to is unapproved, perhaps even untested, by God.  If the lake of fire,² or the outer darkness,³ is the place where the omnipresent God is not, this unapproved mind is as close as a human being can come to it here and now.

Romans 1:28 is the first usage in the New Testament of ἐπιγνώσει (a form of ἐπίγνωσις).  Paul used it two other times in this letter: 1) For no one is declared righteous before him by the works of the law, for through the law comes the knowledge (ἐπίγνωσις) of sin;⁴ and, 2) For I can testify that [my fellow Israelites] are zealous for God, but their zeal is not in line with the truth⁵ (ἐπίγνωσιν, another form of ἐπίγνωσις).  Though Paul doesn’t say it directly I think he has used ἐπίγνωσις⁶ in Greek for the Hebrew word יָדַע (yâdaʽ).

The early history of יָדַע (yâdaʽ) must have been in Paul’s thoughts as he penned Romans.⁷  The first occurrence of יָדַע (yâdaʽ) is from the mouth of the cunning serpent: The serpent said to the woman, “Surely you will not die, for God knows (yâdaʽ, ידע) that when you eat from it⁸ your eyes will open and you will be like divine beings who know (yâdaʽ, ידעי) good and evil.”⁹  Adam and Eve believed the serpent and ate the forbidden fruit.  Then the eyes of both of them opened, and they knew (yâdaʽ, וידעו) they were naked; so they sewed fig leaves together and made coverings for themselves.¹⁰  And the Lord God said, “Now that the man has become like one of us, knowing (yâdaʽ, לדעת) good and evil, he must not be allowed to stretch out his hand and take also from the tree of life and eat, and live forever.”  So the Lord God expelled him from the orchard in Eden to cultivate the ground from which he had been taken.¹¹

It seems to me that Paul must have been thinking about God’s promises through Jeremiah as he wrote Romans 9-11.  I will give them the desire to acknowledge (yâdaʽ, לדעת) that I am the Lord.  I will be their God and they will be my people.  For they will wholeheartedly return to me.¹²  “People will no longer need to teach their neighbors and relatives to know (yâdaʽ, דעו) me.  For all of them, from the least important to the most important, will know (yâdaʽ, ידעו) me,” says the Lord. “For I will forgive their sin and will no longer call to mind the wrong they have done.”¹³  So while the desire for the ידעי (yâdaʽ) of good and evil led to sin and death, the ידעו (yâdaʽ) of God is the way back to righteousness and life.

One more thing about יָדַע (yâdaʽ) bears mentioning and bundles everything together with Paul’s understanding—you also died to the law through the body of Christ, so that you could be joined to another, to the one who was raised from the dead, to bear fruit to God.¹⁴  Now the man had marital relations (yâdaʽ, ידע) with his wife Eve, and she became pregnant and gave birth to Cain.¹⁵  So יָדַע (yâdaʽ) (and ἐπίγνωσις, I think,) describes an intimate form of knowing, much more than a casual acquaintance.

God gave them over to a depraved, reprobate or unapproved (ἀδόκιμον, a form of ἀδόκιμος) mind, and this led inexorably to the social situation we all know intimately (Romans 1:29-31 NET):

They are filled with every kind of unrighteousness,¹⁶ wickedness, covetousness, malice.  They are rife with envy, murder, strife, deceit, hostility.  They are gossips, slanderers, haters of God, insolent, arrogant, boastful, contrivers of all sorts of evil, disobedient to parents, senseless, covenant-breakers, heartless,¹⁷ ruthless.

One final thing should be noted about this unapproved mind.  Left to our own devices we not only practice this worship of unrighteousness, we promote it with missionary zeal.  Although they fully know (ἐπιγνόντες, a form of ἐπιγινώσκω) God’s righteous decree that those who practice such things deserve to die, they not only do them but also approve of those who practice them.¹⁸

Addendum: May 29, 2020
Tables comparing Genesis 3:4; 3:5; 3:7; 3:22; 3:23; Jeremiah 24:7; 31:34; Genesis 4:1; 2:16 and 2:17 in the Tanakh, KJV and NET, and Genesis 3:4; 3:5; 3:7; 3:22; 3:23; Jeremiah 24:7; 31:34 (38:34); Genesis 4:1; 2:16 and 2:17 in the Septuagint (BLB and Elpenor) follow.  Following those are tables comparing Romans 1:29 and 1:31 in the NET and KJV.