I’ve sometimes wondered in the past if the gospels were written after the fact as a way to explain away the end of the temple, Judea, and Jewish culture, but the gospels don’t expound exclusively on the end of the age. They also attest to an age that would arise thereafter, later to be called generally the Christian Age, or Church Age. Even if the gospels were written after the fact, this vision of the future is nonetheless on target; prophetic, even.
The epistles are also prophetic in this way, and we know that at least some of them were written before the age ended. Doubly prophetic. And perhaps the events of the times, namely the First Jewish War, solidified in the disciples’ minds the words of their Christ a few short decades earlier enough to confidently inspire them to memorialize them.
Contrary to the OP, Paul did quote Christ on at least one detail – that of the bread and wine:
For I received from the Lord what I also delivered to you, that the Lord Jesus on the night when he was betrayed took bread, and when he had given thanks, he broke it, and said, “This is my body, which is for you. Do this in remembrance of me.” In the same way also he took the cup, after supper, saying, “This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me.” (1 Cor 11:23-25)
More striking is the well-established existence of James the Just and an unbeliever’s link between him and Jesus:
Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, [or, some of his companions]; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned. (Antiquities 20.9.1.200)
In his epistle, James doesn’t call Jesus his brother, so he may have been a half-brother or a cousin. At any rate, he existed and was martyred sometime between the mid 40s and early 60s AD.