Jesus had a human nature, so wouldn’t he also have had a naturally sinful nature?

Be careful here. One of the blunders many people make is gnosticism (which believes, among other things, that the spiritual and physical are necessarily opposed). This is not so. “Nature” is not corrupt. Corruption is corrupt. Sin is precisely what is contrary to the nature that God calls “good” in the book of Genesis. It is damage to nature, not nature itself, that constitutes sin. Thus, sin (which we all inherit in Adam) is based on a warping and a deformation of our nature. It is never natural for us. Thus Jesus, the perfect man, had a perfect human nature, one that did not include sin.