Judaism doesn't work. It requires a fulfillment of the Mosaic Law. Jesus fulfilled the Mosaic Law at Passover 33 C.E. So Christianity is the legal extension of the Edenic Promise (Genesis 3:15), the Abrahamic Covenant (Genesis 22:18), and the Mosaic Law Covenant (Exodus 19: 5,6).
Jesus was not proselytizing to the Gentiles. Paul (aka, Saul) decided to proselytize to the Gentiles. The question is not whether Judaism works, but whether Christianity is an open club because Paul got his "epiphany."
If it was meant to be an open club, I can only believe Jesus would have told a Roman soldier that he had something to tell him; however, Jesus remained Jewish his entire life. He was trying to reform Judaism from the inside. And bible scholars tend to think of Jesus as an anti-Roman zealot, regardless of the New Testament claim that God Is Love.
Apparently, Christianity was the second time that Romans borrowed a religion to compensate for their seemingly limited spiritual thinking. The first was giving Roman names to Greek pagan gods. The second was adopting the Genesis story (Old Testament) to give credence to becoming a follower of Jesus (originally all Jews).
In my opinion, if one reads the New Testament one reads, "the multitudes rejoiced when Jesus entered Jerusalem." Now who were the multitudes in Jerusalem - yep, Jews. Only Jews. So, again in my opinion, by the time that Christianity was chocked full of Gentiles, one can more honestly say that Jews did not reject Christ, but rejected Gentiles (aka, not wanting to rub elbows with them). In other words, while Jews had to be literate (in the Hebrew writings) to be a Jew, they were now supposed to be co-religionists of a bunch of illiterate ex-pagans? Let's be real!