If that were true, Paul would not quote Psalm 14 because it does not say what you claim. To understand this, one must actually read ALL of Psalm 14. Don't forget to read the last half of the Psalm
also.
Pslam 14
The scoundrel has said in his heart,
"There is no God."
They corrupt, they make loathsome their acts.
There is none who does good.
The LORD from the heavens looked down
on the sons of humankind
to see, is there someone discerning,
someone seeking out God.
All turn astray,
altogether befouled.
There is none who does good.
There is not even one.
Do they not know,
all wrongdoers?
Devourers of my people devoured them like bread.
They did not call the LORD.
There did they sorely fear,
for God is with the righteous band.
In your plot against the poor you are shamed,
for the LORD is his shelter.
Oh, may from Zion come Israel's rescue
when the LORD restores His people's condition.
May Jacob exult,
May Israel rejoice.
In context, it is plain to see that in this Psalm, there is the scoundrel for whom it is said what Paul quotes. And these are set over and against "the righteous band". And the righteous band are composed of Jacob/Israel. That means that gentiles are in view. This makes Paul's case that gentiles have no advantage over Jews.
Other passages quoted by Paul make the case that the enemies are from within, i.e. Israelites. This makes the case that Jews have no advantage. They can be just as evil, in spite of the fact that to them was given the Word of God.
This one Psalm alone wipes out the Calvinist notion that nobody seeks (or has sought) God. The "all" refers to those who have chosen to not believe in the God of Israel. It is they who do no good. It is they who are not discerning. It is the sons of those men who are in view, not the sons of all men which is made obvious by the victimization of "the righteous band" by the scoundrels/fools who say there is no God.
Why would the fool be singled out with a definite article? Because not all men are fools. Otherwise it would read like "all men say, there is no God." By describing a category of men, "fool", the implication that it is not all men.
QED.