As the title states, why are long block lengths commonly assumed or used in channel coding proofs?
Shannon's source coding theorem shows that you can encode at the rate of entropy. Shannon's channel theorem shows that you can transmit data at the rate of channel capacity. In both theorems, both quantities – source entropy and channel capacity – are only approached in the limit of a large block size. If you limit the size of your block, the theorems are not true anymore. The only way to get a clean theory is by allowing the block size to grow unboundedly.