It may be hard wired, but I don’t know that it’s unchangeable, just sometimes very difficult: ”. . . remapping the brain is possible, but for adults it can be a very laborious task, requiring practice every day for years. When I ask Hutchins if there is hope for us poor singers, he laughs. ‘I would say there’s plenty of hope. Practice, practice, practice. A good vocal teacher and patience will help’” (from article linked above.)
My dad says my aunt was “awful” when she started taking singing lessons as a little girl, couldn’t hold a note. Now she can get strangers to cry. She went from one extreme to the other, because she kept at it. I imagine in that way, singing and dance are much alike. Very few people, if any, start out moving gracefully. It takes dancers years of rewiring their physical coordination, but it happens.
An excerpt from a RadioLab episode called “Musical Language” talks about some fascinating aspects of voice, music, and language. Jump to 7:00 for a discussion of tone languages—and how those who speak tone languages seem to have remarkable control over pitch—the first 7 minutes is worth listening to, too, though. They suggest pitch control may be influenced by infant language exposure, but they also point out that means humans have an ability to control pitch, even if it is never developed. So we’ve got the raw materials—not everyone will be a professional singer, but probably everyone can develop a singing voice if they work at it enough.