Let's be fair. The obsession with many (not all!) Christians is not over sin, itself, but what to do about sin. We concern ourselves with sin, because we define sin, not as "doing bad things we shouldn't do," but as "separation from God." Since the crux of Xy is reconciliation with God, separation from God naturally is a concern for us.
But you're right, in a way. We can either spend our time loving God, or we can spend our time hating sin. One pulls us toward God gently. The other sends us running off in some ... direction.
If we spent more time acknowledging our inherent goodness, and celebrating our reconciliation, we would have less hang-ups over sin.