Why can't God persuade all mankind, in their free-will, to believe in him?

The assumption, of course, is that God does exist. A God who is all-powerful and omniscient could certainly convince every person to believe in him, if that is his wish. Yet it is clear that much of mankind does not believe in him, either believing in other gods or no God at all.

Romans 1:19-20 shows that Paul believed in God, but this does not prove to the modern mind that God exists. Francis Bacon seems to have believed that God made an effort, through miracles, to convince idolaters and the superstitious, but thought that no effort was needed to convert atheists (The two books of Francis Bacon: of the proficiency and advancement of learning, Divine and Human, Page 86):

And as concerning divine philosophy or natural theology, it is that knowledge or rudiment knowledge concerning God, which may be obtained by the contemplation of His creatures; which knowledge may be truly termed divine in respect of the object, and natural in respect of the light. The bounds of this knowledge are, that it sufficeth to convince atheism [of God's existence], but not to inform religion: and therefore there was never miracle wrought by God to convert an atheist, because the light of nature might have led him to confess a God: but miracles have been wrought to convert idolaters and the superstitious, because no light of nature extendeth to declare the will and true worship of God.

With the advance of science, it is no longer enough to rely on the cosmological argument (the existence of nature) or the teleological argument (the wonder of nature) to convince atheists of the existence of a Supreme Being (God). Atheists do not believe in God and then reject him - they simply do not see a need for God in their worldview. Francis Bacon says that God made an effort to convince idolaters and the superstitious that he exists, so we can not believe he would not now make the same effort to convince atheists.

As the question states, God is a loving God, and by that virtue, God should be trying hard to convince every person to believe in him. If Francis Bacon is correct, God wants to convince every person to believe in him. The fact that some people cannot be persuaded seems to signify that God is not all-powerful.

Let's consider the answer from a different direction.

Before the flood, early men had more evidence of God's existence. The Bible describes the Garden of Eden being still on the earth and that angels were set to guard the entrance with flaming swords. It would have been hard for anyone to deny the existence of God when the evidence was right before their eyes.

Yet, despite this knowledge, early men did not love or worshiping God more. In our fallen sinful nature, our enmity is against God. Evidence alone, without the softening of our hearts, will only awaken rebellion. This open rejection of God was one of the reasons why their time of mercy ended much sooner. Alas, they were all removed by the flood with the exception of Noah's family.

Since the flood, God in His wisdom has drawn men to Him by the conviction of the heart rather than sight. Even miracles were given in short durations, always with the possibly to doubt. You see, God will never completely remove doubt, but to each of us He gives amble evidence to believe. The Holy Spirit works diligently in everyone's heart, making us realize our need and calling us to repentance. And even when we reject this mercy, judgement does not come to us right away, God continues to work on us until we fully harden our hearts. God also calls many to evangelize, not because He needs us to convict men of the truth, but He wants us to be co-workers in His gospel message, partakers of the the power of the Holy Spirit.

God "desires all men to be saved and to come to the knowledge of truth" (1 Timothy 2:4)

Many do not believe in God because of our own selfish desires. In our pride we do not acknowledge our need for God or see the sinfulness of sin. Even if for a moment we see the possibility of His existence, we do not pursue it, choosing rather the wisdom of the world rather than the foolishness of the cross. I speak this from experience because I was an atheist as long as I remember up until early adulthood. My conversion was solely the power and grace of God, and even my understanding of the theories of science and evolution could not hold me from the simple truth -- we were not created by chance from nothingness, but by the love and personal breath of God.

I think the verses you quote are actually the answer:

What may be known of God is manifest in them for God has shown it to them. For since the creation of the world his invisible attributes are clearly seen, being understood by the things that are made, even his eternal power and Godhead, so that they are without excuse (Romans 1:19,20).

In other words, God already made it obvious in nature that He exists, so why would He waste His time jumping through hoops to prove it to people who simply despise the idea of His existence because they want to live without moral law and are simply recalcitrant in their "unbelief"? Francis Bacon says concerning the idea contained in these verses (The two books of Francis Bacon: of the proficiency and advancement of learning, Divine and Human, Page 86):

And as concerning divine philosophy or natural theology, it is that knowledge or rudiment knowledge concerning God, which may be obtained by the contemplation of His creatures; which knowledge may be truly termed divine in respect of the object, and natural in respect of the light. The bounds of this knowledge are, that it sufficeth to convince atheism [of God's existence], but not to inform religion: and therefore there was never miracle wrought by God to convert an atheist, because the light of nature might have led him to confess a God: but miracles have been wrought to convert idolaters and the superstitious, because no light of nature extendeth to declare the will and true worship of God.

Furthermore, as we consider further the quoted verses, we find that they already know that God exists, for "God is manifest in them." There is no such thing as an atheist, only an anti-theist. Verse 21 shows it even more:

Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened. (Romans 1:21 KJV)

So they knew God, but they rejected Him. Further, it says in verse 25 "They exchanged the truth about God for a lie" (NIV) and one cannot exchange what one never had.

And again, in verse 28 "Furthermore, just as they did not think it worthwhile to retain the knowledge of God, so God gave them over to a depraved mind," (NIV) so it becomes clear, again, that there are no atheists, only anti-theists. They had the knowledge of God's existence, but did not consider it worthwhile to retain, and so they threw it away. Should God now jump through whatever hoops they demand in order to convince them that the knowledge they already have is worthwhile to retain?