The Calvinist theory of preservation of the saints is *not* equivalent to a primary doctrine of the gospel. Teachings of men must be evaluated on their own merits, through scripture. No man-made theory, let alone one that half of Christians do not expouse, is a primary part of the gospel message.
The gospel is simply that the way to God is through faith in Christ. If we believe in our hearts and declare with our mouths that Jesus is Lord, we will be saved.
But let's look at the actual verses:
John 5:24: ""Truly, truly, I say to you, he who
hears My word, and
believes Him who sent Me,
holds eternal life, and does not come into judgment, but has passed out of death into life."
The verbs for hear, believe, and have are all present active participles. That is, one who hears (Continuously) and believes (continually), continues to hold eternal life. 'Passed' is present middle - not a tense we have in English. It isn't referring to a one time past event, but rather to the subject (the person who actively believes) benefitting from the passage from death to life.
1 John 5:13: "I write these things to you who believe in the name of the Son of God so that you may know that you have eternal life."
Again, 'believe' here is a present active participle in the Greek. That is, a literal translation would be 'to those who are continuously believing' in the name....' Hold is also present active.
If we are believing, then we hold eternal life. The passage does not say that those who once believed, but don't now, have eternal life. It doesn't say that a current believer cannot cease to hold eternal life later if he stops believing."
Rom 10:9: "that if you confess with your mouth Jesus as Lord, and believe in your heart that God raised Him from the dead, you will be saved"
Unlike the previous two verses, here confessing and believing are aorist active subjunctives. Aorists aren't that helpful on their own for narrowing down time frames. However, the following verse clarifies (with the use of present passive) that they are both ongoing states, not past events.
"for with the heart a person
believes, resulting in righteousness, and with the mouth he
confesses, resulting in salvation." Rom 10:10
Eph 2:8: "For by grace you have been saved through faith; and that not of yourselves, it is the gift of God;"
'saved' here is another present passive. It is not referring to a punctiliar past event.
Related:
What does it mean that it is by grace we have been saved, through faith, and that this is not of ourselves but is the gift of God?
I could go on and mention many other passages, such as John 3:16, that show the importance of abiding faith (that is, continuous faith) - as only those who believe hold salvation.