In order to answer this question, I feel we must first ask a few other questions. In answering these, I hope it will become evident, even if you disagree, as to why we do not embrace any type of altar call or invitational system in our worship services.
What is the worship service meant for?
Perhaps another way to ask the question is, "Who is the worship service meant for?" In the New Testament we see the "worship services", which include things such as assembling together, fellowship, the breaking of bread (Lord's supper), the preaching and teaching of the Word, and the singing of hymns and spiritual songs, as being a gathering of believers. The Greek word for Church, ekklesia, actually means "the ones called out". Our modern-day "worship services" are meant to be an assembly of "the ones called out" in the worship of our Lord. We are the ones who have access to God, through Christ, and we are the ones who are a "kingdom of priests".
In the Old Testament it was not really any different, in principle. The people of God would gather together to worship. The "worship services" were not used as evangelistic tools to invite outsiders in. Of course, this is not meant to neglect the fact that in every congregation there are likely unbelievers present -be it because they are false professors of the faith or simply "guests". In either case, we are not to tailor the worship of God by His people to those who do not know Him. It is, after all, a "service" of "worship" from God's people to God. It is the people of God serving, lifting up God, being humbled before Him, extolling His greatness, and rejoicing in Him. It is to be about God far more than it is to be about us.
Certainly unbelievers are welcome. We would absolutely never discourage unbelievers to come, and we should pray that the Holy Spirit would use the Word of God in their hearts just as we pray it does in ours -though for their conversion, first.
Does this mean the Gospel is not to be preached in the congregation?
By no means! The Gospel is to be proclaimed regularly, for the Gospel is for Christians as well. The Gospel is not just a doorway for the unbeliever. It is the foundation of the Christian life, and we need to hear it over and over. In Paul's epistle to the Romans, for example, Paul talked about how he wanted to visit the Romans to preach the Gospel to them -to believers! All of this said, it is important that we qualify what we mean by "the Gospel".
In the age we live in, it is very typical to think of the Gospel as a formula. Many of the evangelistic tracts we hand out cater to this idea. We have the "Four Spiritual Laws", for example, and most of the tracts we read, most of the websites we visit, and most of the "invitations" we hear follow that same motif. The Gospel is seen as a bullet-by-bullet list of things you must assent to followed by some activity we perform: either saying a prayer of commitment, coming forward, raising our hand, etc. Let me be the first to say that God draws straight lines through crooked paths and has used those means to expand His kingdom, but we must realize that this is not the way the apostles proclaimed the Gospel, and we must admit that just because God can mercifully use something doesn't mean it is good or right. Some of these formulae are so scant that one seriously wonders if the hearer has any concept of their sin before a holy God, their need for Jesus, and that salvation is by Him alone. A cursory reading of the book of Acts shows us that, while there are common and specific elements among the content of the apostles' preaching, it is not constrained to a disjunct, formulaic checklist to be followed. It was proclaimed and unfolded as news, not a quick pill to be handed out to pagans and swallowed. Likewise, we see in the Scriptures that the only "response" spoken of is the abandonment of our self-salvation and embracing of Christ crucified as our hope, according to God's sure promise. We would do well to be wary of anything that would, even with good intentions, possibly confuse that or add to it.
To illustrate, we are familiar with the fact that some teach the Gospel in terms of "making a commitment to Christ". While it is true that it would be difficult (and un-Biblical) to imagine someone as being a believer and not being committed to Christ, it must also be stated that no man is saved by his commitment. What level of commitment is good enough? Have we committed ourselves enough? Are we sincere enough? This obscures our need for a Savior who has done it all. These invitational systems that call us to perform some activity can truly mislead and result in disillusionment.
Certainly there are many who have believed they are saved just because, in an emotional expression, they made a religious commitment, walked an aisle, prayed a prayer, or cried profusely during a beautiful song about Jesus. To be mislead about the state of our soul is a horrible thing! We have also all heard of people who "tried Christianity". They "got saved" when they were 12 years old, by coming forward at an altar call and making a "decision for Christ", but later rejected Christ after seeing no change and many unmet expectations. Were they really saved? The Bible says "no"; they were mislead about what the Gospel is and how a man is saved. They were lured to something, but it wasn't Christ, and they were lured by something, but it wasn't the Word of God pointing out their need for salvation from sin.
Such ideas, again being well intentioned, confuse the fact that we are saved purely by Christ through embracing His saving work. We need to hear about our real need, Jesus Christ, and God's free promise, not about what we can do. The whole point of the Gospel is that there is nothing we can do to save ourselves. The basic principle is that if we wish to point to something we did, we are not pointing to a solid foundation at all. It is as the great hymn goes,
My hope is built on nothing less
than Jesus' blood and righteousness
I dare not trust the sweetest frame
but wholly lean on Jesus' name.
Now, if a man wishes to express outwardly what has taken place in his heart then that is fine and well -let him be baptized instead!
It is also important to point out that whole practice of "altar calls" is less than 200 years old. The altar call and similar invitational systems became popularized through the revivalist ideas of men like Charles Finney, who believed you could create revival and bring people to faith through the right combination of human ingredients -such as emotional music, sentimentalism, and messages that are meant to produce emotional responses. The idea was that if you can produce what appears to be the proper response through these methods, then they are saved. This is because men like Finney believed that conversion is a natural matter, not a supernatural one. The spiritual fallout from that "revival" was horrible.
Sadly, that belief is still prevalent today. This is seen in such abuses as churches planting people in the crowd who are previously directed to come forward and "prime the pump" during the invitation, making it easier for others coming forward when they see the movement toward the front. Here we again see the idea that if we can humanly produce what seems to be the right response, if we get a count of the number of decisions made, we are saving souls. Are some saved? Certainly. Do the churches mean well? Yes, but neither of these make it right.
We, on the contrary, believe that God uses the preaching of His Word to bring people to Christ by the agency of the Holy Spirit applying it to the heart. We believe that it is not by the eloquence of the preacher or anything we produce (1 Cor 1), but it is purely by God's Word going out and being applied by the Holy Spirit. Conversion is supernatural, and the means God uses in performing it is the proclamation of His truth. There is comfort in that. We can preach the Gospel as clearly as we know how and trust Him for the results rather than create all kinds of human mechanisms and foolishly rely upon them to build Christ's Church. We can send missionaries and trust in God's providence to use their faithfulness, knowing that God ordains the means as well as the ends.
In short, we do not give altar calls and such because not only are they not found in Scripture, but we believe them to be based on un-Biblical premises. A man once asked the great 19th century preacher Charlies Spurgeon why he did not use altar calls and the like. They said that Mr. Spurgeon must "strike while the iron is hot." To this he replied, "If God heats the iron, then it will stay hot." Spurgeon is a profound testimony to the faithfulness of God in using the simple preaching of men to save without the use of human additions.
Then how is it that we preach the Gospel to the congregation if not in this way?
We believe that we do preach the Gospel every week, and we should. How? The Gospel is present through the ministration of God's Word, which, by the use of God's Law, humbles us and reminds us of our need for His mercy, and by the news of Jesus, His Person, and God's grace to us in Him which is found throughout the Bible. Truly, the Bible is all about Christ and our need for Him, so when it is faithfully preached, the Gospel is present. Hence, it is through ordinary preaching that the principles of the Gospel are proclaimed, provided we are rightly dividing the Word of God (for we can all attest that some can preach a sermon on a text of Scripture and leave the hearer in a moralistic stupor rather than humbled before God by being instructed in His beautiful ways and gratefully resting in Christ). Second, the Gospel is present in our corporate prayer and singing of songs. In such things the same elements of being humbled before God, seeing our need for His mercy, the good news of His grace in Christ, and our gratitude toward Him should be expressed if the right attitude is in place and the worship songs are selected carefully. Lastly, the Gospel is present in the sacraments -specifically, the Lord's Supper. The breaking of bread and sharing of the cup is a symbolic restatement of the Gospel, and that is for the enjoyment of believers.
What is the place of evangelism?
The evangelism of unbelievers is certainly to take place within the context of the Church (all believers) through our Lord's commission to us as believers, but not primarily in the worship service. The Body (the Church) can help organize such efforts through local congregations by means of sending missionaries and providing a forum and framework for discipleship, but these things are not to be confused with the people of God assembling to worship God. Likewise, parents are to teach and evangelize their children, and all believers are to go out and teach others about Christ. Keeping evangelism outside of the worship service does not hinder evangelism. It puts it back into its proper place. This is not to say that one cannot be evangelized within the context of a worship service. There is nothing wrong with the preacher addressing the unconverted in the congregation and warning them of their danger outside of Christ, but this does not mean we need to reserve the last 5-10 minutes of the service for the performance of emotion-inducing music and an invitation for people to come forward out of their seats and receive Jesus. If they hear the Word of God, are crushed by their sin, and see Christ as the only Solution, then can they not embrace Him by faith right where they are?
We can restate our reasons for avoiding invitational systems for the following reasons:
1. They are not found in Scripture.
2. They are based upon un-Scriptural beliefs about the purpose of communal worship.
3. They are based upon un-Scriptural beliefs about the nature of conversion.
4. The are based upon the idea that the Gospel is something of a formula or "pill" to be given to pagans.
5. They can obscure and confuse the Gospel, resulting in sinners potentially being misled or disillusioned about what it is to be a Christian and believe in Christ.
6. They are not needed because our God is faithful to use the plain proclamation of His message from His Word to convert sinners.
You are right in insisting that the Gospel should be present during every worship service, but I just hope to gently point you to the fact that the way in which this has been done historically does not include altar calls and the like. Due to the prevalence of thinking today which holds that worship services should be geared toward the unsaved and that a type of invitation/altar call system is necessary for the Gospel to truly be preached, we recognize that we are a minority. However, we believe that our approach is more consistent with what the Scriptures teach in those areas, even if it means that we are unconventional in that regard.
I hope this is sufficient to answer your question, and I hope that you are persuaded according to God's truth. If, according to the Scriptures, you find us in error in this then by all means show us. It is truly welcome, as are questions such as these. It is good that we have a common aim toward His truth, praise God.