Our pastor recently stopped doing an altar-call / invitation at the end of the services. Is this biblical?
My name is Bob Deffinbaugh. I have served as a pastor-teacher and elder at Community Bible Chapel (Richardson, Texas) for nearly 34 years. I grew up in a Baptist church and so I am somewhat familiar with the questions you have raised. I'd like to address your questions on several levels.
It is important to begin by focusing on the primary purposes of the church. The first text I would turn to is Acts 2:42, which tells us that the saints continually devoted themselves to the apostles' teaching, fellowship, the breaking of bread, and prayer. Instruction, worship, fellowship, and prayer are fundamental functions of the church. Note that evangelism is not a stated purpose for the church's gathering. That is because this occurs as the saints go out from the church into a lost world, proclaiming Christ.
A second text would be Ephesians 4:1-16 (especially verses 11-13), which is another central passage of Scripture. The purpose of the church is to edify (build up) believers and bring them to maturity. The purpose of apostles, prophets, evangelists, and pastor-teachers is to equip the saints (the church) for the work of ministry. In that sense it is not the full-time employees of the church who are "in the ministry," as much as those in the churches that they serve.
Somehow it has come about that the primary purpose of the church gathering is assumed to be evangelism. Thus, the pastor preaches the gospel to the lost (who may or may not be there), and then gives an invitation. Rather than training the people to go out and evangelize, the saints are encouraged to invite their unsaved friends to church to hear the gospel. That's pretty much the opposite of what we see in the New Testament.
Therefore, if winning the lost is not the primary purpose for the church gathering on the Lord's Day then one would wonder why it has become a tradition in many churches to preach an evangelistic message and then give an invitation nearly every Sunday. (I think that the gospel should be given in a concise and straightforward way, but that this should not be the primary focus of the message every Sunday.)
It is interesting to me to observe that Jesus never actually gave an evangelistic message (so far as the gospels tell us), nor did He ever give an invitation (as we know it -- see Matthew 11:28-30). When Paul speaks of the conversion of the Ephesians he describes it as "learning Christ" (Ephesians 4:20). Frankly, if we are teaching God's Word as we should people will be learning of Christ, including the lost. I have seen a number of people saved by listening to teaching addressed to believers.
For these reasons I would say that many churches need to quit trying to attract unbelievers to church, seeking to win them to Christ there, but they should start teaching believers the Scriptures, helping saints to know and develop their spiritual gifts, and then sending them out into the world with the gospel.
Now, on another level, I would raise the question of who it is who should lead the church. Of course our Lord Jesus is the Head of the Church (Ephesians 1:22; Colossians 1:18). But the church is to be governed and led by a plurality of elders, and not just one man. (Indeed, there is no office of pastor in the Bible, only the offices of elders and deacons -- 1 Timothy 3; Titus 1; Philippians 1:1. There is the gift of pastor-teacher, as there is the gift of exhortation or giving or helps, but these are not offices.). A decision of this kind should be carefully discussed and studied, and then reached by the elders.
Finally, I would differ not only as to "who" should make such a decision, but "how" it should be enacted. Had the elders reached the decision to place much more emphasis on training up the saints, and setting aside a formal invitation each Sunday (and occasional invitation might not be such a bad thing), then I would hope that the reasons for this decision would be taught to the church from the Scriptures. Indeed, I think it would be good for the teaching to precede the announcement of a decision to make any changes. This way the whole church would see that they are simply seeking to obey the teaching of our Lord.
Below I'll provide some links to a few messages I've given on related texts and subjects.