Modern Judaism and every form of Christianity (like all world religions) make mutually exclusive fact claims...that is each one makes statements of objective truth which contradict the other.
Jesus was either the Jewish Messiah (as Christianity has ALWAYS claimed...from the New Testament onward) or He is not—and hence should be ignored (or even supressed...if He was a blasphemer...). Polite “ecumenicism” is a lie—while we can respect each others tradition—both (modern) Judaism and Christianity cannot be equally correct—if the universe as created by God is coherent.
Therefore it is proper and good for Christians to pray for, and desire the conversion of Jews....and actually, equally proper for Jews to pray for AND EVEN SEEK THE CONVERSION TO JUDAISM of gentile Christians.
After all if Judaism is correct, than Christians are at best, mislead pagans, worshiping a false 3 faced God in a seriously blasphemous way... If Jews love their fellow man—AND they believe Judaism is objectively true, logically they should pray for, and seek the conversion of Christians. SO WHY IS IT THAT JEWS DO NOT PRAY FOR AND SEEK THE CONVERSION TO JUDAISM OF CHRISTIANS?
If however, Christianity is true, and Jews 2,000 years ago and since have rejected and missed their true Messiah, and the One God truly is known in 3 persons...than Jews above even all other persons, need the prayers of Jesus’ followers, that they may know, and love all of ours Messiah as well.
The universal religion for non-Jews is known as ‘The Seven Laws of the Children of Noah’. These are the prohibitions of idolatry, blasphemy, murder, theft, prohibited sexual relations, cruelty to animals, and a command to establish courts of justice. Jews were chosen to bear this message and its details.
The Jewish “Aleinu” prayer does indeed call for the conversion of all the earth to these principles.
Non-Jews need not become Jewish any more than every Christian need be a formal minister or priest.