Why was a man stoned for gathering wood on the Sabbath when Jesus criticized strict Sabbath-keeping?

You have to compare the contexts. In Numbers 15 the striking thing is that the man is only picking up sticks, which is trivial, so the real question is why he is doing it on the Sabbath. He should have gathered his sticks on Friday; if he is gathering them Saturday to burn for heat, that is working. But the elders did not know his motive, whether it was ill intent, mockery, or forgetfulness, so they brought it to Moses, who brought it to God, because in those days they had an infallible interpreter through the Urim and Thummim. Right before this, God had given the law that someone who breaks it inadvertently receives grace and goes to a city of refuge, while someone who does it deliberately is stoned. God, who knows the heart, judged that this man acted deliberately, and so he was stoned. So the passage is about intent, not merely strictness. In Matthew 12, by contrast, Jesus is exposing the Pharisees' hypocrisy. They knew that even on the Sabbath a man may pull his ox out of a ditch, so Jesus draws out the lesson: the Sabbath was made for man, not man for the Sabbath. The Pharisees were not honoring the Sabbath as holy; they were using it to convict others and make themselves look superior. So Numbers 15 and Matthew 12 teach completely different lessons, and in both God is looking into the heart to find the real truth.