The gospels belong to James’s “popularizing” tradition, and James is the so-called “Q source” of the synoptic gospels. Mark gives us a good old fashioned adventure story, and Matthew adds back in more of the wisdom literature stuff. Luke-Acts represents an attempt to “mediate” between the traditions of James-Mark-Matthew and the pseudo-Pauline/Petrine literature.
If this scheme holds, then the old move of contrasting the (nice, liberal) synoptic gospels with (mean old) Paul won’t hold; Paul’s teachings directly inform (but do not exhaustively determine) the Jesus we get in the gospels. We know that Paul knew nothing of Jesus except his death and resurrection, and in this historical scheme, we can’t claim that Paul somehow “removed” the teachings of Jesus — rather, Paul’s minimal historical claims provided the kernel around which the gospels (often called “passion narratives with introductions”) creatively developed.