That misconception is often based on Ezekiel 3:17-19 where God speaks to Ezekiel about being a watchman and warning His people about the coming Babylonian destruction of Jerusalem. God told Ezekiel if he did not give the wicked such a warning, he’d be responsible for their deaths. That passage should never be used in evangelism because it is speaking of a physical destruction, not eternal damnation. That verse is not addressing evangelism.
The Scriptures make it clear that God holds us responsible for contact, not conversion. We can only bring Christ to people; only God can bring people to Christ. Christ testified in John 6:44, “No one can come to me unless the Father who sent me draws him; and I will raise him up at the last day.”