How can we prove to the extremely critical that the people taken for the name of the Lord (Acts 15:14) are the ones referred to in the Scriptures, as the "Bride," "the wife of the Lamb" "The New Jerusalem," etc.?

The “extremely critical” are the very ones to recognize the reasonableness of the doctrine of election so positively outlined in this passage. It is the thinking, critical minds that are rejecting the antiquated, obsolete theories of the past, and now are coming to see the truth concerning the Divine purpose. How reasonable, then, to those who view the facts and reason critically, are the Bible teachings relating to the selection of “a people for His name,” and that this election has been in process of development throughout the entire Christian Era! Naturally, the mind would seek further enlightenment as to the identity of this special people, or class. The various passages of Scripture relating to this class would logically fit into place, and it would be recognized that it is the church of Christ the Bride, the Lamb’s Wife, the “New Jerusalem,” etc. And then how reasonable and consistent the teaching that following this election the Heavenly Father, who is all love and compassion, and goodness, will through this glorified Class–Christ and His Church–rebuild the ruins of David, which are fallen down, restore the Jews to their own land– and “after this,” invite all the Gentiles upon whom His name is called”–the world of mankind–to come back into harmony with Himself and receive eternal life and joy and peace. See the verses following the one cited in the question.