While it is true that sinners cannot have communion with God, it seems to me that our take on this is that, in a sense, we have been kicked out of his presence, like Adam and Eve being kicked out of the Garden. However, this is not how the true narrative reads to me.
First, my understanding of sin is that by definition it is leaving the presence of God; that's what sin is. It is us going our own way. When we sin, we separate ourselves from him. This is reflected in the narrative in that Adam and Eve are not in the presence of God (the One who walks in the Garden), and symbolized by them hiding from Him.
Rather than God "kicking them out," He actually goes and looks for them. This is the God I know. As to our inability to be impure beings in the presence of a holy God, it is still we who created the separation. The analogy that popped into my head was that of a child who left the perfectly clean house and got covered in mud head to toe. The child left the house, but now they can't get back in because they will get the house dirty. They must be cleansed first, hence Jesus's sacrifice.
Again, going back to the narrative. God didn't kick them out of the Garden as a punishment. It was for their own good, so that they wouldn't eat of the tree of life and be stuck that way forever. This matters because it is a much more loving God doing what is necessary for our salvation, and not a punitive God. It also sets up the pattern of God seeking man rather than the other way around.
Am I crazy?