If you ask any rabbi, he will tell you that man does not need faith to be righteous. Righteousness is entirely attained by good works. If you lead a moral and virtuous life you are "acceptable to God." These works have to do with making the "right" choices -- what we know is morally right, and we do know what is morally right by the Commandments.
So, an atheist can be righteous in the eyes of God according to Judaism. Thus, an atheist can be a Jew. A Jew who belives in Christ as His Savior is not.
I believe that you have hit upon the crux of the problem and it is not a matter of whether one follows Judaism, Christianity or paganism. One set will always believe that they can save themselves. The other set will realize that only God can save them. The prophet said that all good works and deeds are as filthy rags. Jesus said who is "good" except the Father? All righteousness, all soul natures come from God. All who hear of God and His ways,and still follow their own notions, no matter what sect they propose to follow, are sinning.