Many believed in God by seeing the miracles performed by Jesus. Today we do not see miracles happening in the same way as the first century, for example. Many disregard the Bible and do not even think it is historical. Although nature and the fulfillment of prophecy etc. prove the existence of God, for many these are not sufficient reasons to accept the existence of God. The fact that a damning fate awaits unbelievers does not seem to have a bearing either. In view of these circumstances I wonder what the optimal approach to preach the Gospel or prove the existence of God?
There is no “optimal approach” to preaching the gospel. I believe that at any time, in any city, town, job place or school, there are a number of “open” people, who are ready to hear the gospel. However, different people will be moved by different things. Some will be motivated by visiting a true New Testament church and seeing the love of the followers of Jesus for one another (John 13: 34-34). For them, you first must be part of a truly vibrant Christian fellowship, and then all you need to do is convince them to come one time. Others will be motivate by your own Christian life–seeing a need to have what you have in Christ (Matthew 5:14-16). For these people, you first must have such a vibrant Christian life, and you need to share your live and convistions. Still others will be motivated by hearing the truth of the gospel and by seeing Jesus (Romans 10:17). For these people, your job is to somehow convince them to simply sit down with you and study the Bible. The Word of God will do most of the work. In this case, I suggest you start with the Book of John (John 20:30-31). Then, there is another group of “open” people who can be convinced by the presentation of evidence such as evidence from science or from fulfillment of prophecy or from a World View argument or from evidence for the historical or archaeological evidence for the Bible. This group can be helped by Christian apologetics. I believe, based on experience, that, of the four groups above, this is the smallest group. More are convinced by seeing our love for one another, or by the live of a disciple of Jesus or by simply reading or hearing the gospel. However, some are convinced by rational arguments using the “evidence” for Christianity.
You also mention the possibility of helping people come to faith because they do not want to go to hell. Some would completely discount this approach–using fear to convert someone. I am not willing to completely discount this. I have personally known people who became Christians, initially, because they feared judgment. There may be some for whom what is needed is for you to warn them about hell.
It is evident, from your question, that you have been using this approach. You have been presenting logical reasons for people to consider being a Christian. One might think that clear, rational. logical arguments would win the day fairly easily because the case for Christianity is so strong. Well…. Most of us humans are more ruled by passion and emotion than by logic and reason. This is true for both the educated and the not so well educated. My suggestion to you is this: You should use all four approaches with people (and maybe even the fifth, sometimes). Do not be discouraged if people do not accept, right away, a great logical argument about biblical prophecy or science and the Bible. This does not necessarily mean that they are not open. If people have an irrational reason for not obeying God, then a rational argument might not change their minds. It is just that arguments are not the means for opening up most people. For most, it is your love, your life, your church, or simply sharing scriptures with them that will be effective. I hope this helps.