The religious teachers of Jesus day said he broke the sabbath. He healed a man and told the man he healed to rise up and carry his bed. So the religious teachers were sure he broke the sabbath and taught others to do so. Would you advise a person to carry their bed on the day that you count as the Sabbath?
Jesus lived in sinless obedience to the Mosaic Law, so he was much more zealous for obedience to it than they were. It is contradictory to believe both that the Pharisees were correct about Jesus breaking the Sabbath by healing on it and that Jesus was correct about it being lawful to heal on the Sabbath. It is lawful to heal on the Sabbath, therefore the Pharisees were incorrect in thinking that Jesus had broken it.
That is true only of Israel. Yahweh commanded Israel to keep Sabbath. Jesus didn't command Christians to keep it. Nor did the apostles. Nor did Paul.
If you want to be a stickler about what was only for Israel, then in Jeremiah 31:33, the New Covenant was only made with the house of Judah and the house of Israel. A Christian is by definition a follower of Christ and Christ followed the Sabbath, so that is what His followers also do. Jesus began his ministry with the Gospel message to repent from our sins for the Kingdom of God is at hand, and the Mosaic Law is how his audience knew what sin is, so repenting from our disobedience to it is an integral part of the Gospel of Christ, including repenting from breaking the Sabbath. Furthermore, Jesus Gentiles are not permitted to sin, sin is defined as the transgression of God's Law, and God's Law commands to keep the Sabbath holy, therefore Gentiles are commanded to keep the Sabbath holy.
In 1 Peter 1:16, we are told to have a holy conduct for God is holy, which is a quote from Leviticus where God was giving instructions for how to have a holy conduct, which straightforwardly includes keeping God's Sabbaths holy (Leviticus 19:2-3). In Colossians 2:16-23, Paul was encouraging the Colossians to continue keeping God's Sabbaths holy in obedience to His commands in accordance with the example that Jesus set for us to follow and to not let any man judge them and keep them from obeying God. In Acts 15:21, it has the expectation that Gentiles would continue to learn about how to obey Moses by hearing him taught every Sabbath in the synagogues, which implies that they were expected to keep the Sabbath holy.
Paul said the Law is like a master disciplining children, enforcing rules, making them obey with threats and punishments. But grace works differently and the Spirit of God is not a spirit of fear nor of enforced obedience but of freedom.
In Psalms 119:29, David wanted God to be gracious to him by teaching him to obey His Law, so that is how God is gracious to us. The word "Torah" means "instruction", though it can be translated as "law" when instructions come with penalties for not following them, however when those penalties our removed by Jesus paying them on our behalf, it still remains instructions for how to walk in God's ways, which all followers of God should seek to do by faith. The fact that Jesus gave himself to pay the penalty for our sins should make us want to go and sin no more, and should not make us want to go back to living in sin.
In Psalms 119:45, God's Law is a law of freedom. The reason why the land of the free is not also the land where there are no laws is because we can abuse our freedoms. So true freedom is not the freedom to do whatever we want, but the freedom to do what we ought. Good legislatures don't create laws with the goal of limiting our freedoms, but with the goal of enhancing them, and no one knows better than God how we ought to live, so we must be like violins who trust God to tie down our strings in order to make music.
The Law cannot make you holy. It didn't make Israel holy. It failed. It was given in part to fail so that no one would boast of their obedience as proof of their worthiness before God. So avoid the Law as proof of personal holiness it isn't what makes a person godly.
God did not fail, but rather the His Law does what it was given to do and does not do what it was not given to, and can't fail for a purpose for which it never given. So we are not made holy by obeying the Law because it was never given for that purpose, but that doesn't mean that we shouldn't obey it for the purposes for which it was given. It is not as thought the fact that we are not made holy by obeying God means that we don't need to obey God.
Nowhere does the Bible say that the Law was given in so that no one would boast of their obedience as proof of their worthiness before God. Abraham was justified by faith long before the Law was given, so God had no need to provide an alternative an unattainable means of becoming justified through obedience to the Law, so it was never given for that purpose. Even if someone manages to live in perfect outward obedience to the Law, then they still wouldn't be justified by their obedience because it was never given for that purpose.
But Jesus healed on the Sabbath, preached on the Sabbath, commanded others to 'work' on the Sabbath, happily observed his own disciples satisfy their hunger by harvesting and threshing on the Sabbath. Sabbatarians know of these examples but do not take them to heart so they make Sabbath keeping a Law for their followers. Jesus didn't do that. He didn't command his followers to keep the Sabbath.
The Sabbath for Christians is Jesus. He is their rest. They rest from works of obedience to the law. They don't run amok breaking every commandment but neither do they measure their godliness by keeping the Law as if it could do them any good whatever.
In Matthew 12:5, Jesus said that priests who performed their duties on the Sabbath were held innocent, so there were some forms of work that were never intended to be understood as being prohibited by the Sabbath, but there are still other forms of work that are prohibited by the Sabbath. Jesus lived in sinless obedience to the Mosaic Law and in Matthew 11:28-30, he was inviting people to become his disciples and to learn from him how to obey the Law, not inviting people to refuse to follow him. By saying that we would find rest for our souls, he remezed Jeremiah 6:16-19, where the Mosaic Law is described as the good way where we will find rest for our souls. This rest for our souls comes from having faith in God to guide us in how to rightly live through His Law.
