One running theme in these stories is that the Pharisees become increasingly uncomfortable with Jesus as things move along. For one thing: he hasn’t been amenable to their authority and they don’t like that. It seems clear that the situation in the synagogue was something of a setup, with Pharisees hanging around, waiting to clutch their pearls in unison. Jesus sees the trap laid out for him, but he can’t in clear conscience walk away from a chance to help a wounded man.

The area of Galilee was not at that time a part of the Roman Empire, but was a “client state”, meaning that they nominally had control over their own laws and trade, but had to pay a tribute to Rome in exchange for getting protection from the legion. Yes, it was a protection racket, and a profitable one for Rome, too. The Herodians were the top family in Galilee since the Maccabean revolt won their sovereignty from the Greeks, and they worked closely with Rome to keep their position.

There is a reasonable argument regarding whether one can do good works on the Sabbath, with some specifically encouraging such activity, while others put strict limits on it. During the time of the Gospel stories, there was a contentious argument within the Pharisee world regarding whether providing medical assistance on the Sabbath was permitted, and this story plays directly into that.

I believe this is also a reference to the battle in which the Jews were slaughtered for refusing to fight on their Sabbath.