My sense about the guy gathering sticks on the Sabbath is that an example was being made. The penalty for breaking the Sabbath had been established as being death (Ex. 31:15). This was the first violation that we are aware of; I think it probable that the letter of the Law was carried out to demonstrate the seriousness with which the Law needed to be taken. We also don't know the extent of the violation. Was the man picking up a few sticks that he needed for an individual fire, or was he doing a day's work, ignoring the Sabbath entirely? Either was prohibited, but the latter might have presented an aggravating factor.
A similar case might be that of Ananias and Sapphira. Clearly, liars are not routinely stricken down by God, but this was a case of a flagrant deception at the very beginning of the Christian church, and an example needed to be made so as to generate a proper fear of violating God's laws.
Throughout the history of Israel, the Law tended to be often forgotten. There were long periods when the festivals such as the Passover and Feast of Tabernacles were not even celebrated. As JWs, we tended to think that the Law was enforced fairly consistently throughout Israel's history, but this was not the case. Times where the Law was consistently kept were few and far between.
By the time of Christ, the Roman Empire had removed the right of the Jews to impose the death penalty for the violation of their laws (John 18:31). Any capital case had to be brought before the Roman authorities. So there would be no executing of anyone for Sabbath-breaking alone.
When Jesus called Himself "Lord of the Sabbath," I don't believe that He was simply saying that He was not subject to the Law; He was saying that He was the God who had given the Sabbath and the other Laws to Israel in the first place. A JW would never read it that way, of course. As such, Jesus was more qualified than anyone else to speak about what the intent of the Law was. And He said, "The Sabbath was made for man, not man for the Sabbath." The Sabbath was never intended as a burdensome religious ritual (as the Pharisees had made it) but as a time of rest and refreshment and of worship. The ritualistic aspect of the Sabbath was not of more importance than human life. As David and his men had eaten the otherwise forbidden showbread in a time of extreme need, it was permissible for Jesus' disciples to pick heads of grain (note - not doing harvest labor, just picking what they needed for their immediate survival). The Pharisees had made nit-picky laws about the Sabbath that would have prohibited even this minor effort, but those man-made laws exceeded what the Law of Moses required. Jesus, as the true authority - the Lawgiver - was within His authority to pronounce what the requirements of the Law actually were. It wasn't that the Law was different now, it was that the Jews had added many detailed ritualistic rules that were never part of the Law to begin with.
As an aside, one familiar with JW teaching might gain some insight on the taking of blood transfusions from this principle regarding the Sabbath.