I'm sure this has been done, but I haven't done it. The "third quest" for the historical Jesus is the re-examination of the historical Jesus that took place after E. P. Sanders had forced a re-examination of Judaism and Paul's writings in the late 70s/early 80s. In 1985 Sanders would move from his re-visioning of Paul to a look at Jesus in his Jesus and Judaism.
Other work taking a second look at Jesus was going on, not least Geza Vermes' work. And there were methodological critiques in play like A. E. Harvey's Jesus and the Constraints of History. N. T. Wright's material on the "third quest" in the chapter he added to Stephen Neill's, The Interpretation of the New Testament, 1861-1986, was very formative for my own understanding.
To some extent, the critiques of earlier quests stand on their own two feet. The earlier quests were too focused on sayings rather than "the contraints of history," which might more centrally lean toward events. N. T. Wright's "double similarity"--the idea that Jesus must fit in between Judaism and earliest Christianity--must be used with caution but is as helpful as "dissimiliarity" was, the idea that we might affirm as historical sayings of Jesus that we can't imagine anyone making up in Judaism or early Christianity.
What I want to do today is ask specifically how the revised perspective on Judaism in the early 80s might have direct or indirect implications on our understanding of the historical Jesus.
1. Jesus was not starting a new religion.
I think this goes without saying. Jesus came for the "lost sheep of Israel" (Matt. 15:24). His time on earth did not include any sense of starting some movement outside of Israel.
2. Jesus' focus was on Israel, not the Gentiles.
The Gospels indicate some interactions between Jesus and non-Jews. We would expect the Gospels, three of which were fashioned for a Gentile context, to highlight those interactions. However, Matthew's emphasis likely gives us the best sense of Jesus' original focus on Jews (Matt. 10:6;15:24). Jesus' mission in Galilee was not about expanding the kingdom beyond Israel but was more about making sure that all Israel would be part of the coming kingdom.
3. Jesus was an apocalyptic prophet.
Those historical reconstructions that make the most sense see Jesus as a prophetic figure against an apocalyptic background. John the Baptist set an apocalyptic stage for Jesus' ministry, and Jesus' central message was the arrival of the kingdom of God. I do think it likely using purely historical criteria that Jesus viewed himself as the Messiah, although I think he avoided this persona in his public ministry. He did not, however, expect the kingdom to come militarily, which sets him apart from revolutionaries like Judas the Galilean.
4. Purity was not a major concern for Jesus.
If John the Baptist gives off some Essene vibes, Jesus doesn't. His message lands on the prophetic side of the "prophets versus priests" tension in the OT. As the prophets repeatedly indicted Israel for thinking their sacrifices would trump their unrighteousness, Jesus strongly endorsed the value of people over Law. We can speculate that Galilee was not particularly focused on purity concerns like those in Jerusalem, and Jesus seems to have followed suit.
I believe this dynamic holds in it the seeds of both why Paul was so annoyed with the Jesus movement and why he took inclusion to the next level after he overcame his resistance. Jesus included the lost sheep into Israel, despite their uncleanness and laxness in relation to the Law. Paul would persecute the early church for this reason and then expand the inclusion to the Gentiles once he had his own experience of Jesus.
Perhaps there is an inner psychological dynamic at work here as well. Paul who had struggled so long with being born in the Diaspora and being thought a second level Jew finally has a cathartic moment of self-acceptance where he embraces both the Diaspora and the whole world.
5. The Pharisees were not the boogie man.
Sanders highlighted the fact that the Pharisees were centered in Jerusalem and that Jesus was actually not likely to have had much interaction with them in Galilee. I take this as a caution rather than as an absolute. I continue to be intrigued by Matthew 23:15.
But the new perspective has highlighted, first, that most Jews actually revered the Pharisees. Not all of them were hypocrites. They were the ones actually trying to keep the covenant. They are the elder brother in the Parable of the Prodigal Son. Some of them became believers without stopping to be Pharisees (Acts 15:5) and Acts has Paul calling himself one in the present tense in Acts 23:6.
Sure, some of them no doubt had their priorities way out of whack--strain out a gnat and swallow a camel. But their attempt to keep the Law was not an attempt to earn salvation. They were born "in." It was no doubt an attempt to gain honor and perhaps some of them dismissed the rest of Israel as being "out" because they weren't keeping the Law as well. But the Mishnah will still say 200 years later that "All Israel has a share in the age to come."
So the Pharisees were strict, but no different from some of my strict relatives who keep a lot of rules. You can keep rules out of devotion to God, which some Pharisees did. Or you can keep rules and make the rules more important than first principles, which undoubtedly some Pharisees did.
6. Thorough expectation of works
I thought of another. While in Paul, I think the focus of his discussion of works has to do with works of Law, that is, those aspects of the Jewish Law that separated Jew from Gentile, Jesus has nothing to say on this debate because he is not dealing with Gentiles or their inclusion in the people of God.
Accordingly, Jesus assumes that God expects a certain kind of righteousness in a person's life. Jesus emphatically does not formulate such works in Pharisaic terms. He does not formulate them in terms of purity concerns or the kinds of "works of Law" that Paul has in mind. For Jesus, these are true works of righteousness, actions that help others (almsgiving) and appropriately honor God (prayer, fasting).
We can retrofit Jesus to be concerned with faith for he is concerned with the heart (e.g., Mark 7). But there is no trace of the faith versus works tension in Jesus. He wants the right works coming from the right heart. To read Paul's debates back on him is anachronistic.