To fulfill the law" means "to cause God's will (as made known in the law) to be obeyed as it should be" (NAS Greek Lexicon: pleroo), so Jesus fulfilled the law by teaching us how to correctly obey it. Our salvation is from sin (Matthew 1:21) and sin is the transgression of God's law (1 John 3:4), so Jesus graciously teaching us to fulfill the law is intrinsically the way that he is giving us his gift of saving us from not fulfilling the law. If Jesus fulfilled the law on behalf believers instead, then that would mean that he would be removing his gift of salvation from believers. Our salvation is about being made to be like Christ, but if we were only counted as being made to be like Christ without actually being made to be like him, then that would be removing our gift of salvation.