Purity in heart is something that no sinner has. "The heart is deceitful above all, and desperately wicked: who can know it? I the Lord search the heart" (Jeremiah 17:9-10 KJV). Further, the prophet added, "O Lord, I know that the way of man is not in himself; it is not in man that walketh to direct his steps. O Lord, correct me, but with judgment; not in thine anger, lest thou bring me to nothing." (10:23-24 KJV)
So, when Jesus tells his disciples that the pure in heart are blessed, for they shall see God, he is pointing them to a future time when that purity of heart shall have been brought about, in the sincerely repentant who agree with God's verdict on their deceitful sinfulness. Note what Ezekiel says on this point:
"...they shall take away all the detestable things thereof and all the
abominations thereof from thence. And I will give them one heart, and
I will put a new spirit within you; and I will take the stony heart
out of their flesh, and will give them an heart of flesh...
Repent and turn from all your transgressions; so iniquity shall not be
your ruin. Cast away from you all your transgressions, whereby ye have
transgressed; and make you a new heart and a new spirit: for why will
ye die, O house of Israel? For I have no pleasure in the death of him
that dieth, saith the Lord God: wherefore turn and live ye." (Ezekiel
11:18-19 & 18:30-32 KJV)
Jesus and his disciples knew well such Hebrew scriptures. Further, John the Baptist had first come, to prepare the way of the Lord, and his message was of repentance with water baptism. All those who submitted to that were prepared to then receive the Messiah and his gospel. Repentance is absolutely requisite for God to deal with our sin. It is not doubt that causes sin, but sin that is not repented of that blocks saving faith.
Jesus was stating that purity of heart would be the portion of those who would see God. Such ones would turn, and live. Repenting is the turning, which God enables in those who have been prepared to receive the good seed of the kingdom. It then takes root in a cleansed heart and produces fruit.
Unrepentant sinners are certainly full of doubts, but those willingly unrepentant are full of resentment against God, bitterness, and self-righteousness. Unrepentant sinners may come to a point where they face up to their sin and the need to repent, if they see that the gospel is the power of God unto salvation to all who believe, that in it is revealed the righteousness of God; that the wrath of God shall be "revealed from heaven against all ungodliness and unrighteousness of men, who hold the truth in unrighteousness" - Romans 1:16-18. But the willingly unrepentant will never have pure hearts and they will never see God.
It is not doubt that causes sin, but sin that is not repented of that blocks saving faith. The willingly unrepentant are defying God, raising their angry fists against him and saying that he is unjust and unrighteous. Your quote from Christopher Hitchens is a classic demonstration of that.