"Edgar, how does one account for the words 'not lawful' in Matt. 12:4 and 'profane' or 'desecrate' in Matt. 12:5, if Jesus was in fact arguing that His disciples were not breaking the sabbath, instead of arguing that He has the authority to dispense them from the obligation of keeping the sabbath?"
I do not believe that Jesus is agreeing with his opponents, who charge the disciples with practicing what is "unlawful" on the Sabbath (Matthew 12:1-2). The Lord is replying to the baseless aspersions of the Pharisees who suggest that what the disciples of Jesus are doing on the Sabbath is unlawful or forbidden. But Christ uses the examples of David and the priests who work on the Sabbath to prove that God allows those engaged in his work to do what could be viewed as "unlawful" or could be seen as technically profaning the Sabbath. However, notice that Christ says the priests who work on the Sabbath are blameless (Matthew 12:5). A fortiori, why should not Christ's disciples also be considered blameless while they carry out God's work and pluck grain from the fields through which they walk (Matthew 12:6)?
There are some interesting observations found on this account in Aquinas' Catena Aurea. John Chrysostom states that the disciples broke the Sabbath law, but not "absolutely." They were given an "out," so to speak, because they were hungry. Jerome thinks that the disciples "broke the letter of the Sabbath," but he appears to believe that the charge of the Pharisees was false. Jerome writes:
"But the laws of God are never contrary one to another; wisely therefore, wherein His disciples might be accused of having transgressed them, He shews that therein they followed the examples of Achimelech [sic] and David; and this their pretended charge of breaking the sabbath He retorts truly, and not having the plea of necessity, upon those who had brought the accusation."
John Nolland (The Gospel of Matthew: A Commentary on the Greek Text) notes that the first mention of the Sabbath in Matthew's Gospel is found at our text in 12:1. After providing commentary that centers on what ancient Jewish writings (including the Philonic texts) stated about the Sabbath, Nolland then writes:
"a sympathetic viewpoint on the situation of the needy is likely to have treated their [the disciples'] eating in the fields on the sabbath as not constituting work that would violate the sabbath. Such was not the view of Philo or of the Pharisees we meet in the Gospels. But it clearly is the view of Jesus" (p. 482).
Commenting on Matthew 12:5, Nolland also maintains:
"Again what is established in that the non-work requirement of the sabbath is not absolute . . . Once more at best the comparison creates a space in which apparently unlawful behavior may be justified on other grounds" (p. 484).