Thank you for the interpretative question, “Is Matthew 16:25 teaching the way to eternal life?”
As one reads the larger context of Matthew 16:25, the text speaks about discipleship, not having or earning eternal life.
Matthew 16 says this, “21 From that time on Jesus began to explain to his disciples that he must go to Jerusalem and suffer many things at the hands of the elders, the chief priests and the teachers of the law, and that he must be killed and on the third day be raised to life. 22 Peter took him aside and began to rebuke him. “Never, Lord!” he said. “This shall never happen to you!” 23 Jesus turned and said to Peter, “Get behind me, Satan! You are a stumbling block to me; you do not have in mind the concerns of God, but merely human concerns.”
24 Then Jesus said to his disciples, “Whoever wants to be my disciple must deny themselves and take up their cross and follow me. 25 For whoever wants to save their life will lose it, but whoever loses their life for me will find it. 26 What good will it be for someone to gain the whole world, yet forfeit their soul? Or what can anyone give in exchange for their soul? 27 For the Son of Man is going to come in his Father’s glory with his angels, and then he will reward each person according to what they have done. 28 “Truly I tell you, some who are standing here will not taste death before they see the Son of Man coming in his kingdom” (A searchable online Bible in over 150 versions and 50 languages).
As one reads verse 21 and 24, Jesus is teaching his disciples: “21 From that time on Jesus began to explain to his disciples that he must go to Jerusalem and suffer many things at the hands of the elders, the chief priests and the teachers of the law, and that he must be killed and on the third day be raised to life…24 Then Jesus said to his disciples, “Whoever wants to be my disciple must deny themselves and take up their cross and follow me.”
Three times the words, his/my disciples are mentioned in the context. Thus Jesus is addressing his followers of the cost of following Jesus. Jesus was willing to obey the Father in going to the cross and if they wanted to be his followers they too must take take up their cross and followhis example of obedience to God. It is costly to follow Jesus.
The disciples already placed their faith in him as the Messiah of Israel, but now they were to learn that he will be the suffering Messiah for the sins of the whole world. As John the Baptist declared that he is the Lamb of God who takes away the sin of the world (John 1:29).
As one ponders verses 25–26, there are two major interpretation. The first is that an individual loses one’s salvation or a person loses one reward. Does a person loses one’s salvation if one doesn’t take up one’s cross and follow Jesus? I do not believe so since salvation is by grace through apart from works (Eph. 2:8–9). Salvation is given to whoever calls upon the name of the Lord, not called upon one’s good works (Romans 10::9–10).
If Jesus refused to go the cross, he would have his kingdom of God, but there wouldn’t be no humans in it. All the human will be condemned and apart from the kingdom of God.
If a person spent his/her gaining everything physical in this world, it is unlikely that his/her soul would be spiritual enriched. S/he would die as a rich and famous person but be spiritual impoverished before God. Would that person be able to exchange the riches of this world for heavenly treasures? One can’t buy heavenly riches with earthly riches. That person will know that s/he has lived a selfish life on earth and did nothing to advance the kingdom of God. All of one’s riches can’t buy another soul for God or even for his own soul. That person soul’s is purchased by the blood of Jesus Christ. Nothing can substitute for it.
Jesus is presenting an argument that if a person can’t purchase his soul with earthly riches, why is one spending a life time in pursuing earthly riches when one will leave it all behind. It is foolishness. As followers of Jesus, the pursuit of earthly riches is vanity. His followers are to avoid this foolish path.
As an encouragement for his followers, Jesus states that he is returning with the angels and his reward is with him. To those who carry their cross and follow Jesus, they will receive rewards from him. To those who refuse to carry their cross and to follow Jesus, their works on earth will be wood, hay and stubble instead of gold, silver, and precious stone for those who follow Jesus (I Cor. 3:10–15).
SUMMARY: Salvation is by grace through faith but rewards are given by counting the cost and following Jesus.
For more perspectives:
https://www.quora.com/Is-Matthew-16-25-teaching-the-way-to-eternal-life