Interesting take as usual. Where do you learn this stuff? It is not biblical. You might be surprised to know that in their commentaries Jews show that they understand that they are not able to attain righteousness on their own, it must be imparted to them from the Lord.
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.