God doesn't banish people because of their sin. He banishes them because they reject the Holy Spirit. That's the only sin that keeps them out of heaven. What does happen with sinners is that they lose out on his blessings here on earth because they walk their own way. They don't see the Kingdom of God on this earth because they won't obey and walk the way God wants them to. If they repent and turn away from their sin, they can once again be blessed by the God who does love them. He doesn't want anyone to perish but because he is a just God, he allows them to when they refuse his Holy Spirit.
Job 31:2-3
2For what is the allotment of God from above,
And the inheritance of the Almighty from on high?
3Is it not destruction for the wicked,
And disaster for the workers of iniquity?
Psalm 5:5
The boastful shall not stand in Your sight; You hate all workers of iniquity.
Psalm 7:13-15
13He also prepares for Himself instruments of death;
He makes His arrows into fiery shafts.
14Behold, the wicked brings forth iniquity;
Yes, he conceives trouble and brings forth falsehood.
15He made a pit and dug it out,
And has fallen into the ditch which he made.
The "pit" and "ditch" are metaphors for hell. The wicked are indeed banished for their iniquity. Please learn to read the Bible is the context of it's entirety.
He banishes them because they reject the Holy Spirit.
No, that is the only sin that cannot be forgiven. But all workers of iniquity reject the witness of the Holy Spirit.
John 3:
19This is the verdict: Light has come into the world, but men loved darkness instead of light because their deeds were evil. 20Everyone who does evil hates the light, and will not come into the light for fear that his deeds will be exposed. 21But whoever lives by the truth comes into the light, so that it may be seen plainly that what he has done has been done through God.
What does happen with sinners is that they lose out on his blessings here on earth because they walk their own way.
No, what happens is that the wicked are cast into the Lake of Fire where there is weeping and gnashing of teeth.
Matthew 13:41-43
41The Son of Man will send out His angels, and they will gather out of His kingdom all things that offend, and those who practice lawlessness, 42and will cast them into the furnace of fire. There will be wailing and gnashing of teeth. 43Then the righteous will shine forth as the sun in the kingdom of their Father. He who has ears to hear, let him hear!
They don't see the Kingdom of God on this earth because they won't obey and walk the way God wants them to.
They don't see because they are blind and love the darkness.
If they repent and turn away from their sin, they can once again be blessed by the God who does love them.
No one can come to the Son unless the Father draws them.
John 6:43-45
43Jesus therefore answered and said to them, "Do not murmur among yourselves. 44No one can come to Me unless the Father who sent Me draws him; and I will raise him up at the last day. 45It is written in the prophets, "And they shall all be taught by God.'[a] Therefore everyone who has heard and learned[b] from the Father comes to Me.
Romans 8:
5Those who live according to the sinful nature have their minds set on what that nature desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires. 6The mind of sinful man[e] is death, but the mind controlled by the Spirit is life and peace; 7the sinful mind[f] is hostile to God. It does not submit to God's law, nor can it do so. 8Those controlled by the sinful nature cannot please God.
Those who have not been regenerated by the Spirit of God cannot please God in any way, shape or form, they are dead spiritually and cannot repent unless God first enables them.