Ok. However there is the small matter of Jesus being identified in the Old Testament, combined with Josephus 100 or so years after Jesus' death (nevermind the new testament)...Come to think of it, that may not necessarily be right, entirely. We do have Josephus's Antiquities, after all, helter-skelter as its textual history may be. And although it would be in very distorted form, we'd also have some inkling of Jesus in the exegeses of the Church Fathers, each with their individual axe to grind, granted, together with the tiny handful of additional Roman chronicles out there, in addition to Josephus. These may not give anywhere near as vivid a picture of Jesus as the Scriptural texts do, but they do address your conundrum: we would indeed know Jesus from a hole in the ground. He would not have disappeared from the historical record. We would simply know him as a historical human being who inspired some unusual philosophical activity, although we might well be mystified as to how come such a fuss was made over his followers during much of the first millennium b.c.e.<shrug>
As an afterthought, after 1945, we would also have a somewhat sharper picture of him as well, having found the non-Biblical Gospel of Thomas at Nag Hammadi.
Get's rather complicated eh?