There are several things to consider when reading texts in the New Testament that refer to obedience to the Mosaic Law and Christians.
1. The Jewish Christians of the First Century did obey the Mosaic Law. As Acts 21.17-26 shows, the Jerusalem Church and Jewish Christians in general observed the Law, and even Paul did. In fact St. Paul’s public action in these verses was to prove that anything Paul taught about not being obligated to follow the Law was not applicable to Jewish Christians. Peter, for example, had never eaten anything but kosher food as Acts 10.9-14 shows even though Christ’s earthly ministry was completed and behind him. While most of the epistles are written to Gentiles (i.e., to the Romans, Corinthians, Galatians, Ephesians, etc.) some are not (James, for instance). Therefore one needs to be careful that they know the audience and time period of each letter before they jump to conclusions regarding what the exact meaning of the words in reference to the “law” mean.
2. Jews did not and still do not have a doctrine of original sin nor believe that obedience to the Mosaic Law grants “salvation.” It should be noted that St. Paul repeatedly argued with Christians who wanted to make Torah obedience a requisite for Christian living upon Gentiles. Some of these were apparently Gentiles themselves who were attracted to the customs and lifestyle of the Hebrews. But being that the Gospel does not require any ethnic group to discard their customs and adopt another before one can gain salvation in Christ, Paul taught that such a demand was unreasonable. It should be of interest that St. Paul uses the Mosaic Law, quoting from it repeatedly in his letter to the Romans to prove that the Law does not promise salvation in exchange for obedience to it. In other words, Paul’s teaching is that forcing the Law on Gentiles as a requisite for salvation actually causes the person to break the Law, making following the Law a moot point.
3. Sometimes the “law” spoken of in Scripture refers to being obedient to Christ and sometimes it is referring to the “law of sin and death.” And these expressions are not always clear as to which is being referred to. Mistakes regarding which law is being addressed are made by readers.–See Romans chapter 7.
By the time the Christian Bible was canonized in the fourth century the majority of Christians were now Gentiles. Many expressions in the New Testament did not apply to them as they had been addressed to Jewish Christians who still found freedom in being Torah obedient. (Matthew 5.17-19) Distrust and hatred for the Jews as a whole grew after the sixth century when the Catholic Jewish population and Church in Jerusalem dissolved. And a gravely mistaken misinterpretation of Judaism and obedience to Law grew out of the arguments of the Reformation as Luther used the Pharisees as an “evil characterization” of the Catholic clergy. This warped and cemented anti-Semitism into Christian theology for a while, both Protestant and Catholic, and led to the pogroms, persecutions, expelling of the Jewish from Spain in 1492, and eventually fueled the Third Reich in its “answer to the Jewish problem.”
The Catholic Church has since reversed many of these mistaken views and removed them from its theology. Today Catholics of Hebrew ethnicity may freely observe Jewish customs based on the Mosaic Law, but such observance is not viewed as obligatory or a requisite for salvation. Gentiles, the Church teaches, were never under the Law to begin with, so they are not “bound by law.”
Finally the expression “bound by law” originates with Romans 7.1-3 and refers not being “bound” to the Mosaic Law but “bound” to one’s spouse in marriage. It is used as an illustration of being bound to the laws of sin and death and any requirement of law, Mosaic or otherwise, that stands over a person while they are alive. Scholars view that it is a mistake to think Paul was speaking of the Mosaic Law exclusively in this case because he wrote: “We are discharged from the law, dead to that which held us captive.” The Romans were never charged to keep the Mosaic Law. Even in death Jews are buried under procedures which stem from the Mosaic Law (so they are bound even after death). And the Mosaic Law was viewed by the Israelites as the opposite of Egyptian captivity, so it is not likely that this is the law “which held us captive.”