"Why don't the epistles discuss the things Jesus said and did during his earthly ministry?" is the question that most keeps me on the fence about the existence of a historical Jesus.
One of answers I sometimes get is that the epistles dealt with issues that they arose after Jesus' death. Therefore, the things Jesus said and did during his earthly ministry weren't relevant to the letter writers' purposes. For example, questions regarding the administration and practices of early Christian communities wouldn't have come up during Jesus' earthly ministry so his teachings would not have been helpful in addressing the issues facing Paul.
I find this unconvincing because it doesn't square for me with what I see the early church doing. When an issue arose that Paul hadn't dealt with somewhere in one of his letters, people simply forged new letters in Paul's name in order to make it seem as though he had dealt with it. If Jesus' teachings were accepted as authoritative, I would expect that people would have invented teachings to make it seem that Jesus had addressed problems that he never addressed as well. It might have been difficult to pass off a letter as being written by Jesus, but conveniently remembering what he said as necessary wouldn't have been hard.
Perhaps Paul had too many scruples to invent things that Jesus said, but I can't imagine that the people who forged letters in Paul's name would have had any such qualms. Moreover, the guys who wrote letters attributed to Peter surely wouldn't have hesitated to include things that Peter supposedly remembered Jesus saying if they understood him to have been Jesus' closest disciple.