Why were the Gospels written so long after the events they describe if they’re truly integral to salvation?

Perhaps a general answer would be similar to Peter’s words in 2 Peter 1:12-15:

…I intend always to remind you of these…, though you know them and are established in the truth that you have. {13} I think it right, as long as I am in this body, to stir you up by way of reminder, {14}  since I know that the putting off of my body will be soon, as our Lord Jesus Christ made clear to me. {15} And I will make every effort so that after my departure you may be able at any time to recall these things.

Note particularly v15: Peter says he’s writing his letter so that, even after his death, they will remember the things he is teaching. It could be that the Gospels were written down some time after the events they describe because the original eyewitnesses (especially the inspired ones [John 14:26]) of the events the Gospels describe were reaching the end of their lives and would soon no longer be able to orally verify that accounts of Jesus’ life  given in teaching and preaching were true (cf. 1 Cor. 15:1-11; Luke 1:1-4).

Before the Gospels were written, the gospel message was still known because people preached it (e.g., Mat. 28:19-20). The eyewitnesses who were alive at the time could verify and vouch for the validity of the message that was being preached (cf. 1 Cor. 15:1-11; 1 John 1:1-4).

We can see the relationship between the initial preaching  and the later writing of the Gospels in the opening to Luke’s Gospel, which says:

Inasmuch as many have undertaken to compile a narrative of the things that have been accomplished among us, {2} just as those who from the beginning were eyewitnesses and ministers of the word have delivered them to us, {3} it seemed good to me also, having followed all things closely for some time past, to write an orderly account for you, most excellent Theophilus, {4} that you may have certainty concerning the things you have been taught.

Luke’s Gospel, it says, was written particularly to give a man called Theophilus ‘certainty concerning the things [he had] been taught’ (Luke 1:4). Prior to the Gospel being written Theophilus had already been taught the gospel message (v4); the Gospel was written as a record of eyewitness accounts (v2-3) to verify the things Theophilus had been taught (v4).