Well, I think that "the Christ" is an inner potential that is latent in all men and women. When a particular person, such as Jesus, Buddha, Ramakrishna, Gandhi, White Buffalo Woman, Rama, and many others...when a person like that appears somewhere and profoundly impresses many people at the time, it sometimes results in a new religion being formed afterward as a result. In the further development of the new religion embellishments are inevitably added to the tale, and so it goes. That's how myths arise over a long period of time. I think they all have real events at their core, but people add further stuff of their own invention on top of the story of those events, and the story keeps changing.

What you describe the Christian church doing sounds accurate enough to me.