I would say the core of the story is that Jesus was a man but was also the Son of God and that he gave his life for mankind.
Please note that at least (the underlined) half of this "core" falls out of the field of "history" into that of theology, by definition (btw, the "for mankind" part would not be that easy to demonstrate on purely exegetical grounds).
Ah I thought you elided the general point that nearly everyone accepts some parts of the gospels are embellished for the far grander claim that it would not matter if most or all of it were in fact historically inaccurate. It is one thing for many liberal believers to concede that much of the gospels is myth; it would be quite another for them to abandon the basic story about Jesus as the saviour sent by God to give his life altogether.
As far as liberals are concerned, I really don't think so: theological liberalism, from the 19th-century onward, has been more than keen on discarding the providential and redemptive aspect of Jesus' death, in order to retain Jesus as a moralist, making his death a mere socio-political consequence of his preaching (à la Socrates), devoid of any supranatural efficiency.
Anyway, in my last post I only intended to clarify my initial statement, which you apparently misinterpreted (you can read it again on the previous page): namely that Christianity spiritually feeds on all its source texts, whatever their connection to history. The texts which "nearly everyone" recognise as fictions are not dropped out of the canon, nor out of the church services, inasmuch as they keep on playing their religious role. From this perspective the rest of the debate is quantitative: how much is historical? The diehard fundies would say 100 %, the mythicists 0 %, the Jesus Seminar 17 %... pick your figure, but the most important question is actually qualitative: how relevant to Christianity are the x % retained? More than one century ago Albert Schweitzer came out with the embarrassing conclusion that the historical Jesus (whom he believed had existed) would be a total stranger to the modern Christian faith. Should the historical quest end up with such a character, Christianity would have no choice but "kill him again" to return to its foundational texts...