It is going to vary from person to person. We cannot do what Jesus did. Jesus knew all things about everyone. That is an unfair advantage. Jesus could speak things about someone's life and know exactly what problems needed to be addressed, and that got the attention of the subject. Only a god could know those things. We cannot do that. All we can do is give an answer. If they do not listen to you and me, they will be reached by someone they can relate to at some point, as the spirit moves them.
Consider children. As children, we humans do not have the frontal lobe fully developed until we are early to mid 20s. That is when and where the intellectual decision making becomes (choke) fully developed. A child can understand right and wrong, is heavily convicted by conscience, until they turn a deaf ear and the conscience is seared for good. Children understand rewards and are driven by obtaining them. Children understand consequences. This is the reason most children in the USA who accept Jesus, do so by the time they are 18. If they do not, they usually won't without some other major influence (catastrophe, children, etc, etc). Those are statistics from Barna. This is not to say God does not save adults. He does. But most of the people accepting Jesus have traditionally been children under 18.
Some people are intellectuals and can be reached by debate, apologetics. A Case For Christ would reach those people. Some people are not intellectual people, maybe not very educated, but they are saved by the love, kindness, etc of a church family or rehabilitation ministry. I work in lots of churches, para church ministries and charities. I see all of this. I see and know people who are not educated at ALL. But they know the bible and they appear to be saved. Some of them have poor black churches on the rough side of KC Kansas. They can't get deep into theological debates on minutia, but they are leading their flocks to the best of their ability and God's gifting. The call of God is broad and has to reach people where they are, in whatever state they are in, with the faculties and languages they utilize. The bible, etc. The bible says that men will be judged by their conscience. So, I think appealing to the conscience is well within bounds. The error is saying God CAN'T use a certain technique. Paul acknowledged that people might preach in ways we do not agree with and might have a self-serving motive. He didn't care and said it is all good because the Gospel is being preached and Truth is the Truth regardless. God will use it.