Apologetics is to win people mainly, not just to win arguments.
Hmmm... I thought it was to give an account of the hope that is within. The Spirit can use that to either bring someone to Christ or silence the mouth of fools or even convince a thick headed brother like me of the truth of some point of doctrine.
All apologetical arguments are like tools in your workshed. A hammer will work for some people, a screwdriver for others. I am not sure why presups and evidentialists always have infighting. Why diss the other one's tools?
Now I'm going to go all technical on you as a math teacher. Apologetics is at the root a system of logic. Systems of logic have to follow the rules of logic. For deductive reasoning to work, there are two laws, the law of syllogism and the law of detachment. Any logical argumentation starts with axioms (presuppositions) undefined terms, definitions, and properties -- from those basic blocks, theorems (doctrine in theology) are built. Even an existentialist has presuppositions ... if we don't have presuppositions, we don't have logic. A presuppositional apologist is just puts the axioms in a box and says these are the things upon which I build my system.
Really, there is no truth that is undeniable - because people will deny even the most basic truths.
My pastor tells a story of a woman that brought her husband to a psychiatrist and told him that her husband thought he was dead. She asks him if he can help. He says yes, then takes a lancet (secretly) from a counter and the follow conversation follows....
Psychiatrist: Are you dead.
Husband: Yes, I died about 2 months ago.
Psychiatrist: Do dead men bleed?
Husband: No, they don't bleed.
The psychiatrist takes the lancet, pokes the man in the hand at which point the man starts bleeding.
Psychiatrist: See, you're bleeding, you were wrong!
Husband in astonished exclamation: Doctor, you are right; I was wrong. Dead men do bleed!
However, many folks find a fascination with complicated apologetics and spend much time on it.
Because apologetics is one of the best means of sharpening our understanding of doctrine. What is doctrine, but the teachings of the Bible? What is the teachings of the Bible but the things revealed that are for us and for our children?
Most people come to Christ not for logical reasons alone anyway, but for sentimental reasons and by being loved into the kingdom, or coming to God in a crisis or having close friends and family that have become Christians and have had changed lives.
No one is "argued" into the kingdom of God. No one is converted for sentimental reasons. The only way anyone comes into the Kingdom is through the work of the Holy Spirit, drawing them to Christ, changing their heart of stone to a heart of flesh. Once that is done, they will come to faith and believe. It is not the method that matters, it is the Spirit that matters.
My experience with some of those who love apologetics is that they read much and actually use their apologetics little and, when they do, they do so by going after the jugular and are after a "win" in their syllogisms and not after helping win the person to Christ.
Hopefully, those that are "going for the jugular" are doing so because someone is blaspheming God and they are refuting the statements of fools. Not all of our interaction with those outside the Kingdom is to bring someone into the Kingdom ...
If a terrorist breaks into the church and starts shooting, the interaction ought to be designed to stop the terrorist as quickly as possible with regard to protecting the flock from the wolves. That interaction has nothing to do with winning them to Christ. It has everything to do with protecting the innocent. If we see what amounts to a spiritual attack on the flock, I would expect the same level of care for defending the innocent should be used. During the attack, you don't worry about the person attacking, you defend the innocent. We honor those in the military that defend physical life, we should also honor those in the "spiritual military" that defend spiritual life.
The most effective witnessing I have seen has always been done by folks who are largely ignorant of apologetics? Why is this so?
I would suspect that the cases you have seen are ones in which the Holy Spirit, by the council of his will, chose that time to harvest the person being witnessed to by the mouth of the babe. It is not always so ... and I imagine that in the majority of the cases where true faith has occurred, the person has been under the preaching of the word. (Rom 10:14) The ordinary means of grace (being raised in a covenant family, hearing the word preached week by week, and never knowing a time when the covenant child did not believe) almost always uses those that are strong in apologetics (the pastor).
Also, I have noticed that most people that read deeply in Reformed apologetics are already Christians. Nonbelievers prefer the layman oriented books like Josh McDowell's stuff? Or C.S. Lewis' Mere Christianity approach, rather than Van Til.
My wife came to Christ through reading Francis Schaeffer. Might not be the norm, but then the norm is those within the church. Ultimately, it is the Holy Spirit that works within the heart. What we do is not effectively call, that is the Holy Spirit.
What am I missing? Why don't I like Reformed apologetics? Am I wrong for taking Josh McDowell over Van Til any day?
Nothing wrong with preferring McDowell over Van Til. Both are human authors, both have their place. What I see is that you appear to look at everything in the light of "how does this help me be a better evangelist" which probably means you are good and it. Those that are good at defending the faith might be looking at everything through "how does this help me better understand the God I worship" and they would be very good at it. The only way to have that become wrong is if you started thinking that the whole body should be an evangelist (just like an apologist would be wrong to think the whole body should be made of apologists). We all have a duty to be able to give an account of the hope within; we all need to be able to defend the faith; we all need to be able to plant seed; we all need to be able to cultivate the garden. Celebrate the diversity of the body of Christ ... we are not all evangelists; we are not all teachers; we are not all workers of mercy.
How is apologetics and missions/evangelism related? How do most people put it into action?
They are all part of the work of the church of Christ. Emphasis differs between differing parts of the body. The body cannot function without all the differing parts. While we all need to be able to do any role when pressed, we all have a role that we are best at.