Christ was born with the same physically depraved body and natural propensities all men are born with. He was made like unto His brethren in every way. "IN A SENSE", this physical depravity of the sensibilities could be rightfully termed, a proclivity to sin or even a 'sinful nature.' Bear in mind that when I say a sinful nature, I am NOT saying Christ had sin, for there is a clear distinction between being born with a proclivity to sin, or a nature that provides great temptation to sin which serves as an influence to or a temptation to sin, and NOT sin itself. A 'sinful nature' as I have described it is again NOT sin in and of itself. It is merely the physical depravity in all humans subsequent to and as a result of the fall of man. Such a nature serves as a formidable influence to sin but is not sin until as James clearly points out, the will of man makes a choice to form intents with such depraved influences.
Such a depraved nature is indeed a primary source of all temptation. Scripture is clear that Christ was "tempted in ALL points as we are, yet without sin." If Christ was not born with a depraved physical propensities, such a clear statement as Scripture makes concerning the temptations of Christ being in 'every way' like His brethren, He simply could not have been tempted as we are.
Being born with a proclivity to sin, or even 'a sinful nature' in the sense I described, does NOT necessitate sin in anyone, especially Christ. Depraved sensibilities, again, serve only as a formidable influence to sin, and does NOT necessitate sin itself.