The Scripture is telling us that He chooes not to. He has revealed Himself to all men (Titus 2:11)
Dead men cannot choose life - it is given to them
Dead men cannot sin either! With life comes freedom and with freedom the freedom of the will, so that we can come to the Father willingly (not on our own will!), and live a life for which we will deserve a judgment.
God inscribed Himself in our hearts. Not just the knowledge of Him, but the knoweldge of love, and justice, of mercy and of forgiveness. These are His gifts to humanity. What we do with them is our record for which we shall be judged.
>>He has revealed Himself to all men<<
Pauls letter is to Titus, a fellow believer. Verses 11&12, when put in context with the rest of the chapter refers to saved Christians, not gentiles.
>>With life comes freedom and with freedom the freedom of the will, so that we can come to the Father willingly (not on our own will!),<<
So, which is it? Willingly, or not on our own will? You've contradicted yourself.
We are all dead in our tresspasses, and only though the gift of salvation can we become alive in Christ.
Eph 2:1-10 -
And you were dead in the trespasses and sins in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience-- among whom we all once lived in the passions of our flesh, carrying out the desires of the body and the mind, and were by nature children of wrath, like the rest of mankind. But God, being rich in mercy, because of the great love with which he loved us, even when we were dead in our trespasses, made us alive together with Christ--by grace you have been saved-- and raised us up with him and seated us with him in the heavenly places in Christ Jesus, so that in the coming ages he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus. For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, not a result of works, so that no one may boast. For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them."
Did you catch that? He MADE US alive in Christ - not "you chose to be" alive in Christ. All grace, no works. Your concept of "free will" is largely a myth perpetuated by those who would seek to hamper God's utter sovereignty and authority over His creation.