None of Jesus' actions nor statements about the sabbath meant that He was breaking, improving on, or replacing the Law of the sabbath. Rather, Jesus' statements about the sabbath explained the Law of the sabbath.
Jesus was ministering to Jews who were still under the Old Covenant, and His statements need to be understood in that context.
The manner in which the spiritual meaning of the sabbath finds fulfillment in the lives of New Testament believers is another subject. But the point in the Book of Matthew and in this particular Blog post is that Jesus refuted the Pharisees' wrong application of the Law of the sabbath by explaining, upholding, obeying, and fulfilling the true meaning of the law of the sabbath - and at the time, He expected His disciples also to do no less.
Nothing Jesus ever said or did broke the Law or improved on the Law. He exemplified Law-keeping and taught Law-keeping on every point, exactly as it was originally intended.
2 ...which was not lawful to do upon the sabbath day.
Only in the Pharisees' minds was it unlawful to pick corn on the sabbath day under these circumstances.
Jesus explained that it was "lawful" - i.e., it was consistent with the law of the sabbath - to do good on the sabbath day - e.g., by healing the sick or lifting a sheep that has fallen into a pit.
Jesus said the Pharisees were condemning the guiltless, i.e., He was saying that His discipls were in fact guiltless under Moses' Law.
In order to explain the true meaning of the Law in regard to the sabbath, Jesus appealed to three sections of the Scriptures: He appealed to the writings of Moses in the Law itself; He illustrated it by an incident in the life of David; and He backed it up with a Scripture from the prophets - plus He added an appeal to a commonsense behaviour which even the Pharisees themselves were doing.
Jesus showed in the Law that the laws of the sabbath were never meant to over-arch the sacredness of other important functions in the house of God. Even under the Law, there were certain things which a person could do on the sabbath day without incurring guilt (verse 5).
An incident in the life of David illustrated the principle that some laws - such as those governing the use of the shewbread in the Temple - were not necessarily enforced in moments of life-and-death emergency.
The prophet illustrated the spirit in which the Law was intended, by saying, "I will have mercy and not sacrifice".
On top of all that, the Pharisees themselves had enough commonsense to lift a sheep from a pit on the sabbath day. Therefore they were actually being hypocritical.
Seeing then that the Law itself, when understood correctly, vindicated the disciples who were picking corn on the sabbath day - how much were they vindicated seeing a greater than the Temple was with them!
The Pharisees were misapplying the Law and hypocritically condemning the guiltless. But the truth under the Law always was that the sabbath was made for man and not man for the sabbath. That was not some new idea which Jesus was introducing - Jesus was merely explaining the truth as it always was under the Law.
The son of man is Lord of the sabbath - without breaking the Law! In fact, the Law itself - and principles in the Law - upholds the Lordship of the son of man over the sabbath!
The Law itself allowed that the Temple and its required functions could over-arch certain requirements of the law pertaining to the sabbath - how much more, now that a greater than the Temple was here! This put the laws of the sabbaths and the disciples' actions that day, in proper perspective.