Such an approach is the result of muddled and unbiblical thinking. For a start, when the Bible uses the word 'heart' it does not imply the seat of the emotions but rather the depths of the personality which include the intellect. We are, after all, called to love God with all our 'heart, mind, soul and strength', ie the totality of our beings.
Secondly, one of the words most commonly used to describe Paul at work is that he 'reasoned' with his audience (Acts 17:2,17; 18:4,19; 19:8,9; 24:25). The Greek word used here is 'dialegomai' which is the root of our word 'dialogue', implying a two-way exchange of views.
Other descriptions of Paul at work paint a fuller picture. We learn that he was 'explaining, proving and persuading' (Acts 17:2-4), 'disputing' (Acts 17:18), 'trying to persuade' (Acts 18:4) and 'arguing persuasively about the kingdom of God' (Acts 19:8). He had 'daily discussions in the lecture hall' (Acts 19:9), affirmed his message was 'true and reasonable' (Acts 26:25) and 'tried to convince' the Jews in Rome (Acts 28:23).
Large numbers were 'convinced' (Acts 19:26), and even King Agrippa was 'almost persuaded' (Acts 26:28). Indeed, Paul summed up his ministry with the phrase 'we try to persuade men' (2 Cor 5:11).
Apollos we read 'vigorously refuted the Jews in public debate, proving from the Scriptures that Jesus was the Christ'. Philip climbed alongside the Ethiopian to answer his questions, and again and again the apostles supported their testimony by the evidence of witnesses to the events they proclaimed.
What then of Paul's approach at Athens (Acts 17:22ff)? Luke goes into great detail in this sermon and there is absolutely nothing in the text to suggest he did this as an example of how not to do it. We must assume this was written for our instruction. Certainly, only a few people were converted, but the fact that some were must say something about the orthodoxy of his message. Neither was he dealing here with Jews who believed the Scriptures and the promises concerning the Messiah. These people were miles away in their thinking - the wonder is that anyone was converted! Acts 18:5 described Paul subsequently changing the way he spent his time, not his method of evangelism. He no longer needed to spend his time tentmaking. Neither does 1 Cor 2:1-5 imply a different approach after Athens. At both places his message was thought to be foolish. He did not depend on oratory or compromise his message with the prevailing wisdom. He proclaimed the powerful message of the Gospel, and laid bare the revealed wisdom of God. At Athens they 'sneered' (Acts 17:32).
A detailed examination of Paul's address to the Athenians shows that even though Paul was here engaged in a monologue, it had all the hallmarks of a dialogue. He had evidently listened very carefully to what the Athenians believed, and made his case, not by appealing to the Scriptures which he believed but they knew nothing about, but from the writers, poets and philosophers that were familiar to them. There was not a single Bible quotation recorded in his talk to these 'non-Bible' people, yet what he said was entirely biblical.
We have a shorter account in Acts 14 of Paul addressing another non-Jewish audience, the farmers at Lystra. Here he argued from nature rather than philosophy but again did not quote the Scriptures. We have no other detailed account of Paul addressing non-Jews.
From these examples, we have to work out our approach to people who don't believe the Bible. Unlike Paul's audiences, many of the people we speak to do not even believe in God. We cannot properly begin to address them until we have first listened to where they are in their thinking and unbelief. When we know what they believe and understand, then we can attempt to explain the Gospel to them. In dialogue we can constantly review whether we are communicating. They will say what they don't understand or feel they cannot believe. We can argue our cause in the light of their understanding. We will see if our arguments stand up or whether the concepts of the Gospel are being understood.
Faith, we are told, comes through hearing (Rom 10:14). It has been well said that it is not what we say that matters but what they hear. In the dialogue situation we can constantly review the all-important matter of whether they are hearing us. In God's sovereign wisdom this is his chosen way of bringing people to salvation - through hearing the Gospel. It is the Gospel itself that is the powerful agent at work (Rom 1:16) - that is why it is so important not only that we go to great pains to preach it faithfully but we do our best to make sure we are being heard. Having heard it and understood what we are saying, some will dismiss it as 'foolishness'; but others, whom God in his sovereign purposes has called, will fall on their knees and declare it to be the wisdom of God (1 Cor 1:24). He alone can open their eyes to see it; we cannot do that for them (Matt 11:25-27). Our task is to enable them to hear.
Neither is this some novel or sub-Christian approach. It is supremely Christian. As Jesus was sent into our world, so we also are sent out into the world for others. The late Michael Ramsay commented that we must 'go out and put ourselves with loving sympathy inside the doubts of the doubting, the questions of the questioners and the loneliness of those who have lost the way'. It is there, as we make ourselves vulnerable and give ourselves humbly to people, that we have the possibility of engaging in authentic and powerful evangelism.
When people listen to a monologue presentation of the Gospel, they come up with all sorts of objections in their minds. If we don't give them a chance to come back to us, how will we know what's stopping them believing? Similarly if we know nothing about the person we are talking to, how are we to communicate the Gospel relevantly, in a way they understand? Both Jesus and Paul dialogued with their hearers, and so should we.