John R. Rice believed and taught and preached that we are in a partnership with God. What the OP is missing entirely is that we must have God's power or nothing will happen. Evangelism is useless in our own strength. We go out to win souls, but we cannot do it without God's power, and He is the One who actually saves the soul. Here are statements from John R. Rice to the effect that it is God who saves souls, from the book The Soul-Winner's Fire, probably the one referred to in the OP:
"Every Christian who is willing to do it in God's way and in God's power may have the marvelous privilege of winning souls" (p. iv).
About Ps. 126: "The heathen were impressed with the marvelous deliverance which God gave His people. Real, Heaven-sent revivals are like that. Outsiders know when God has visited His people. A great moving of repentance among the people of God, of tears, or restitution and reformation, times of joy and blessing and reconciliation, make a profound impression upon a community round about" (p. 23).
"We must realize that soul winning is not done in human power and human wisdom. it takes the power of the Holy Spirit to convict and to convert a sinner" (p. 27).
"Unless there is a supernatural change of heart, a regeneration, a new birth, the sinner is still a lost sinner, a rejecter of Christ, a rebel against God, justly condemned and Hell-bound. Admit it now, dear soul winner, you must have heavenly help in this business of saving sinners" (p. 37).
There are many similar statements in this and JRR's other books, which the writer of the OP could have found if he tried, and thus avoided the false accusation of the OP.