would a reverse image search work if the only other place that image exists is on somebody's private social media account?

Depends on if the reverse image service can crawl the private social media account. So, if by private you mean is not accessible and never has been accessible - then probably not. But as they say on the internet, if it's on the internet assume it's public and anyone can see it.

No way of telling unless you try it

I've done searches on images I found on facebook or twitter just to see if I could connect to something ... it seems to work some times. Many people still leave their social media accounts wide open ... some accounts are left open to fish people into their content ... others are left open because people don't know any better. However, this was true a few years ago and less and less so now as more and more social media accounts become locked down ... not so much for security reasons but to keep the content contained to that social media system.

No. As the image indexer wouldn't of had chance to index it.