There are two main points to address why this argumentation is fallacious:
1) It completely ignores historical precedent. Some of the greatest and most renowned evangelists and missionaries have been Calvinists. For example, John Eliot, an early Puritan in New England, was a missionary and evangelist to the Native Americans, as was the 18th century David Brainard. Brainard himself was a friend of the famous Jonathan Edwards, whose evangelical efforts led to the First Great Awakening, and who later served a missionary role among the Native Americans himself. On top of this, we have William Carey, George Whitefield, Charles Spurgeon, David Livingstone and many, many others. Anyone who thinks evangelism and missionary work is an issue for the devout Christian who adheres to the doctrines of grace is simply making bluster.
2) It is based on a fallacious presupposition. The argument seems to come from the thinking that if God elects people, then He can only do it by one way, which is direct divine intervention akin to the apostle Paul on the road to Damascus. This is simply not the case: while the Good Shepherd knows who His sheep are, He is able to ordain the means by which the sheep are called into His flock.
I would like to briefly look at a biblical passage to address this:
The next day He purposed to go into Galilee, and He found Philip. And Jesus said to him, "Follow Me." Now Philip was from Bethsaida, of the city of Andrew and Peter. Philip found Nathanael and said to him, "We have found Him of whom Moses in the Law and also the Prophets wrote--Jesus of Nazareth, the son of Joseph."The first thing to note is the finding of Philip by Christ: Philip did not seek Christ nor was it Philip who found Christ, but rather it was Christ who found Philip. Upon finding him, Christ simply says: "Follow me." This was the same command given to Matthew at his tax collection booth (Matt 9:9; Mark 2:14; Luke 5:27), from which Matthew immediately rose up and followed Christ. Philip likewise is so overcome by Christ's simply command that he runs to his brother Nathanael and tells him that they have found the Messiah.
Nathanael said to him, "Can any good thing come out of Nazareth?"
Philip said to him, "Come and see."
Jesus saw Nathanael coming to Him, and said of him, "Behold, an Israelite indeed, in whom there is no deceit!"
Nathanael said to Him, "How do You know me?"
Jesus answered and said to him, "Before Philip called you, when you were under the fig tree, I saw you."
Nathanael answered Him, "Rabbi, You are the Son of God; You are the King of Israel." [John 1:43-49; NASB]
Nathanael, for his part, is doubtful, but nonetheless goes to see Christ for himself. Christ greets Nathanael with a statement regarding his character and ethnicity, which takes Nathanael by surprise - how could this Man from Nazareth whom he had never met before know about him? Christ replies, "Before Philip called you, when you were under the fig tree, I saw you."
When Christ said "I saw you," the root word for what is translated as "I saw" (εἰδῶ) actually means knowing, and is often translated "see" in a figurative sense. This could be compared to the English phrase, "I see what you mean," which could likewise mean, "I know what you mean." Christ is not merely saying that He saw Nathanael under the tree, like a psychic might have a vision of someone being at work, but rather Christ knew Nathanael before Philip came to call him to Christ. This is why, upon coming to Him, Christ greets Nathanael with, "Behold, an Israelite indeed, in whom there is no deceit!" Christ had His finger on Nathanael before Philip even arrived to the tree.
Indeed, Christ knew who among all of Israel would be chosen as His apostles, and their selection was not by their own merits or actions but by His own will. This is made clear later on in John's gospel when the Lord says, "You did not choose Me but I chose you, and appointed you that you would go and bear fruit..." (John 15:16). Both Nathanael and Philip were chosen by Christ to be His disciples, and there was no possibility that either could have ended up as Judas.
Here comes the connection with this post's topic: God is the chief designer in the means by which the sheep are called into the flock. Philip was effectually called directly by Christ, yet Christ likewise used Philip to bring Nathanael to Him. Christ could have easily gone to Nathanael and called him as directly as He called Philip, yet it was His will that Philip be used to bring Nathanael to Him. In a similar manner is missions: we are commanded by Christ to make disciples of all nations (Matt 28:19), and we will be used by God those other sheep of His flock to Him.
Note on this last point: Christ had commanded the apostles to go out into the world and preach the gospel, and yet He had stated, "I have other sheep, which are not of this fold; I must bring them also, and they will hear My voice; and they will become one flock with one shepherd" (John 10:16). On the one hand, the action is given to the apostles; on the other hand, Christ states that He is the active party in the bringing together of the flock. Is there a contradiction? Not at all. Christ is the active party, but - just as He brought Nathanael to Himself using Philip - He uses missionary work as one means to bring together those "other sheep" into the fold.
God is sovereign over all, but, contrary to what hyper-Calvinists would believe, He has a use for we broken vessels. This is why we must endeavor to evangelize to the lost while at the same time remembering that they are never saved because of us, only in spite of us.