I think there can be Christianity even if we don't believe in a mythologised Joshua ben Joseph, and there can be Buddhism even if we don't believe in a mythologised Siddhartha Gotama.
Mythologised stories have a practiced, theatrical feel to them, and that's certainly the case with the stories of Buddha. Like the tales told of the Zoroastrian saviour
Saoshyant, and his mythological successor Joshua ben Joseph, the stories of Siddhartha Gotama speak more of divine birthright than secular wisdom gained by trial and error. All three saviours are Campbellian in scope and grandeur, their lives full of carefully-constructed tests to demonstrate their unparallelled sagacity, power and charisma.
But just as Christianity would be a different thing if it didn't depend on the mythological truth of Jesus, I think that Buddhism would be a different thing if it didn't rely on the mythological truth of Buddha. In each case, I think each teaching would be more accountable for its thought, more restrained in its claims, and perhaps more open to other ideas. Naturally, given my secular humanistic perspective, I'd consider that an improvement.