Last night, I gave a talk on the Temptation of Christ at an event for the DePaul Humanities Center. My focus was on the devil’s offer to give Christ all the kingdoms of this world, which I used as a starting point for tracing the roots of apocalyptic in the Jewish tradition and its legacy in Christianity. One point I emphasized was the intertwining of religious and political concerns in the second and third temptations, which Matthew and Luke present in a different order. I suggested that this differing order might indicate a certain interchangeability between the political and the religious.
Whenever I make claims like this, the angel of biblical scholarship always appears on my shoulder and starts saying things like: “You don’t know that Matthew and Luke knew of each other’s work! They could have both been reworking a shared oral tradition, etc., etc.” And on my other shoulder, an angel of not-caring appears, saying, “You know what? We can’t know any of that for sure. All we know for sure is the texts we actually have in front of us, and they certainly give an impression of being in rivalry with each other!”
Finally, forced to mediate between these two influences — and I’ll let you decide which is good and which is evil — I decide that the angel of biblical scholarship is right to advise caution, but that the angel of not-caring is ultimately pointing us toward something that is more interesting. I don’t have to make a hard historical claim that Luke read Matthew and decided he could do better, but it’s more interesting than hand-waving toward an “oral tradition.” It’s more discussable if you start from the premise that Luke took a look at Matthew’s Nativity narrative and said, “Wow, this is a hot mess — it needs a serious rewrite!”
I would go so far as to extend this to the Gospel of John as well. I know, I know — it probably represents an independent oral tradition (Ockham’s principle of ontological parsimony is not among the axioms of biblical scholarship), which furthermore was written by a community rather than a single author, etc., etc. People have made good arguments for that, and far be it from me to presume that I could overthrow the scholarly consensus. But again, we can’t know any of that for sure, and treating John as an attempt to “correct” the synoptics is just more interesting and discussable in my opinion.
This tendency to favor the angel of not-caring might stem from my experience in the Great Books classroom, where my goal is less to ensure students follow disciplinary guidelines and more to get them engaged with the texts as directly as possible. In that setting, gesturing toward some hypothetical entity like an “oral tradition,” which by definition we can never have access to, is a distraction from our work on the text that we actually have in front of us. When someone swoops in with that kind of #actually, experience shows that conversation gets shut down. Better to start with the bold, or even over-bold, hypothesis, which at least gets us digging through the text itself.
Actually, it would likely be more accurate to say that my pre-existing impatience with disciplinarity made a Great Books setting congenial to me — and classroom experience affirmed my intuitions in this regard.
An und für sich 
A principle of most controversy? J/k – you didn’t indicate a maximal approach – and I would enjoy a class that highlights controversial edges and requires repeated close looks.
Me, I look for where the real magic is (for my particular consciousness and its likely resonances). Usually I end up “missing the point” – I can’t remember much of the Bible at all, except for some formulations that have really stuck with me. In a classroom setting, that would lead to a quite tedious, subjective, “here’s what I personally value” roundtable, which would have no academic value at all.
I would have certainly learned more through your methodology.
This “[i]n that setting, gesturing toward some hypothetical entity like an “oral tradition,” which by definition we can never have access to, is a distraction from our work on the text that we actually have in front of us. When someone swoops in with that kind of #actually, experience shows that conversation gets shut down. Better to start with the bold, or even over-bold, hypothesis, which at least gets us digging through the text itself.” Sounds akin to the Anthropic principle in Physics: “the reason that things are the way they are is that if they weren’t we wouldn’t be here to wonder about them”. Not satisfying scientifically, but we really can’t be in a universe where physics operated with different laws. While appealing to Oral Tradition might not be satisying for a scholar, we have only what we have in front of us, and no way to know other wise.
Not only do I think you’re right that that you should be able to formulate a hypothesis and see where it gets you in terms of reading a text, but I’m betting there are NT scholars who advocate what you’re arguing. It seems like those detractors are just annoyed you’re not pointing to their model of Gospel relationship (Matthew and Luke use Mark but they don’t know each other; John doesn’t care about the Synoptics). Having been trained in Hebrew Bible and now moving to Comparative Literature, I can sympathize with your frustrations (having been innoculated with that kind of disciplinarity). At a certain point our ways of reading ancient texts is conjectural, and it even seems at times fruitless to give a big explanation about why one advocates for a certain relationship with the Gospels (or in my case, scholars talking about the Pentateuch), since those same people will disagree with you no matter how in depth you go.
As a NT scholar, this is almost *exactly* how I approach all of these texts both historically and literarily. Frankly, “oral tradition” is often a canard used to discipline how people read these texts, preventing any reading that strays too far from orthodoxy.