Why can't God persuade all mankind, in their free-will, to believe in him? Jesus, before He was risen to heaven, left the task of evangelism to his disciples and Christians.  Since men are fallible and are not all-powerful, they are taught to pray and rely on God in their evangelical work.  The question is, since God is all-powerful and omniscient, why can't God persuade all mankind, in their free-will, to believe in him? 

According to Romans 1:19-20, God tries to reveal himself to his people. 


  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).


However, God is also a loving God, and by that virtue, God should be trying hard to convince every person to believe in him.  The fact that some people cannot be persuaded seems to signify either that God is not all-powerful, or He does not want to.

(Note: this question assumes this is possible without conflicting with mankind's free-will, as God can assess each individual's knowledge/personality/character and convince them through multiple mediums such as people/nature/experiences).
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?