ANSWER: You are correct in saying that “many think they are saved but they are not” and these people will no doubt “be very upset they were not taken” when the Rapture occurs. Yet they will learn that it was their rejection of God’s salvation that prevented them from being taken and not because they were not chosen.
In Matthew 22:1-14 the Lord Yahushua told a parable to illustrate God’s invitation to sinners to come and partake of His “gospel feast.” Verse 4 says, “He sent other servants, saying, Tell them which are bidden, Behold, I have prepared my dinner, my oxen and my fatlings are killed, and all things are ready, come unto the marriage.” In this beautiful picture we see God, in love and grace, providing salvation for all men. The oxen and the fatlings being killed speak of the cost to God to provide this salvation, for they speak of Christ’s death on the cross where He willingly took the sinner’s place to pay the penalty of our sins. After being judged for our sins He cried out, in John 19:30, “It is finished.” He finished the work to atone for our sins and because of this God can say to the sinner, “all things are ready, come unto the marriage.” But as we read through the parable we see those who were invited refusing to come to the feast God had prepared. Verses 11-13 are especially solemn, for we read of one in verse 11 who “came in hither not having a wedding garment,” and because of this he was, according to verse 13, “cast into outer darkness.” This man represents those who you mentioned that “think they are saved but they are not.” They make a profession of faith, but because they have refused to accept God’s salvation on His terms they are rejected by God. If they had accepted Christ as their Savior they would have been clothed “with the garments of salvation” as brought out in Isaiah 61:10.
It is true that those who are saved were also chosen. But in our presentation of the gospel to sinners the emphasis should be on God’s love in providing a Savior and their responsibility to accept the Lord Yahushua Mashiach/Christ by faith. John 3:16 declares “God so loved the world, that He gave His only begotten Son, that whosoever believeth in Him should not perish, but have everlasting life.” This is the message sinners need to hear and to believe. Notice it doesn’t say, “God so loved the chosen,” but that “God so loved the world.” God loves ALL MEN and if they refuse to accept His salvation it will be their fault alone when the Rapture occurs and they are left behind.
The truth we’ve been considering is also pictured in the parable of the Ten Virgins in Matthew 25:1-13. In verse 1 we see that all ten virgins “took their lamps, and went forth to meet the Bridegroom.” They represent all who make a profession of faith in Christ and are waiting for the Bridegroom to come. But in verse 2 we learn that “five of them were wise, and five of them were foolish.” The wise virgins picture true believers who have accepted Christ as their Savior and possess the Holy Spirit, and the other five picture mere professors who are void of the Spirit. The “oil” in verses 3-4 speaks of the power the Holy Spirit gives to shine as a light for Christ. We see in verse 10 that when the bridegroom came “they that were ready went in with him to the marriage: and the door was shut.” Those who had trusted Christ were raptured to glory, but the door of grace was shut on those who merely professed to be saved. How solemn to see their reaction to this and the Lord’s response to them in verses 11-12, “Afterward came also the other virgins, saying, Lord, Lord, open to us. But He answered and said, Verily I say unto you, I know you not.” They are surprised and perhaps upset that they were not taken, thinking that they had been saved and were ready, but the Lord gives them the tragic news, “I know you not.” Notice again, He doesn’t say “I CHOSE you not,” but rather “I know you not.” If only they had heard Christ’s voice in the gospel and accepted Him as their Savior, they would have been known of Christ, as we see in John 10:27-28, “My sheep hear my voice, and I know them, and they follow me; and I give unto them eternal life.” But they had chosen to trust in their own good works instead of the finished work of Christ and thus they will be left behind when the Bridegroom returns. In Matthew 7:22 we learn they will plead their good works before Christ, “Many will say to Me in that day, Lord, Lord, have we not prophesied in Thy name? and in Thy name have cast out devils? and in Thy name done many wonderful works?” But their self-righteous pleadings will be met with these words, “I never knew you: depart from me, ye that work iniquity.” They will learn then that it was THEIR CHOICE to reject God’s salvation and continue in disobedience that seals their doom, and not because God had not chosen them.