You're making a kind of category error, partly just by projecting your concept of history and record onto the past, partly by being seemingly unaware of the prevalence of oral tradition in the time and culture.
For the latter bit, the culture surrounding first century Judaism was at least partially an oral one. Portions of the religious tradition were probably exclusively transmitted orally. If you take later Rabbinic claims at face value - which they maintain today - there was a continuous oral tradition of things which not only were not written down but were forbidden to be written down. But these oral traditions were no less authoritative than the written ones. We can see this in the New Testament where trained Pharisees such as St Stephen or St Paul freely refer to oral traditions without qualification, and seem to expect their audiences to accept them as authoritative.
People also interacted with texts in a really different way. Writing something down was an expensive proposition, which is why the act of doing so was a kind of enshrinement or monument to the importance of the thing. The word for "scripture" in Greek literally just means things written. The implication being something along the lines of things
important enough to be written. At the same time, it was normal for people to memorize huge amounts of scripture. Memorization of the entire Psalter was common. So common that church canons actually technically still require a bishop to have it memorized. Young Jewish males who were being formally trained would have the Torah memorized by age twelve. St Paul almost certainly had the entirety of the Torah and Psalter memorized, probably even the entire OT as we know it. St John Chrysostom is said to have had the entire OT memorized. But this isn't limited to religious studies - many ancient sources talk about the importance of memorization and mnemonic devices for memorization. It's just a completely different framework.
In a culture where oral memory is shared, this sharing is in community as well. So repeated stories are shared by the community collectively, and by individuals specifically, in a public way. Just like if fifteen people have a song memorized and someone sings the wrong word others can correct them, so oral societies can self-regulate for a high degree of sophistication and accuracy. But the focus is very different than how we handle modern history. We know that human memories can become "archival" memories of events, and cann be incredibly
stable and repeatable. We as a literate society in a long tradition of literacy tend to over-emphasize the accuracy and stability of writing, and underappreciate the same in oral tradition (fascinating article
here on this).
As for direct quotes... these just weren't common in classical history. They weren't a thing, and they weren't expected. Historical records of famous speeches are something like post-facto expanded footnotes. If you heard a speech, and a historian asked you, the historian would take your account and use rhetoric to recreate what might have been said. This is almost certainly what St Luke has done in his gospel. While this may move away from the idea of a direct quote, there's no reason at all it has to move away from what was communicated. If certain teachings were often repeated, and the gospels are full of them to the point that some are echoed by St Paul nearly word for word, there's no reason these teachings wouldn't be faithfully remembered. We still do this "my teacher used to always say..." The gospels depict Jesus with a fairly consistent and repeated message, summarized as "repent for the kingdom of heaven is at hand." The Sermon on the Mount or other discourses are probably amplified recreations of things Jesus might have said on any given day, and probably contain things he did say. But again, this is normal for how history was done. They had a different focus, intent, and expectation in their writing that we do.
What seems haphazard to you would probably have been seen by the Apostles
and their congregations as combination of unnecessary and useless. It isn't coincidental that the gospels began to be written right after the Apostles generation begins to age out.
There's a really good book that talks about a lot of this by Richard Bauckham called Jesus and the Eyewitnesses.