No, it wasn’t the Jews who believed that circumcision was necessary.
Paul was a Jew, and obviously he didn’t feel it was necessary for salvation.
In fact, the original Jewish Christians were divided on the circumcision requirement (Acts 15) and the Jerusalem council made a final decision regarding what was necessary. Perhaps you don’t understand this chapter so well, so I’ll explain. I don’t believe that anyone in the Jewish Christian community thought circumcision was necessary FOR SALVATION. But they thought that once a Gentile was saved through faith, they had to obey the entire Mosaic Law (when one was circumcised, they were putting themselves in subjection to the entire law). For a new Gentile believer, this would be a hard load, the opposite of what Jesus said his burden was. So, the council decided to limit the requirements for a new believer to those mentioned in Acts 15, which were the points that would allow them to fellowship with the circumcised Jews in Christ. Since that was decided after Jesus (on the authority of the Apostles) it stands to reason that it should still apply today. Moreover, it was decided with the assumption that the new believers would “learn as they went” about the Torah (without which, it is difficult to understand the New Covenant) by going to synagogue.
But it’s wrong to say that the Jews thought circumcision was necessary for salvation. They didn’t, for a Gentile.
I’d like you understand this: if one is circumcised, they are agreeing to be bound by the law of Moses. That applies to you. Paul was emphasizing (in light of the controversy which still seemed to be going on, cf. Acts 15) that Gentile (Roman) believers did not need to follow the Mosaic Law to be saved; therefore, they should become circumcised and put themselves under that Law.
Correct. We have no disagreement here. However, God wants his people to be holy. His standard of holiness is outlined in the Torah. The sign of obedience to the Torah is circumcision. Just because it is not necessary does not mean it is bad; Paul does say that the Jew has a benefit.