Is there a technical reason "why", when I do a rename in MC and the folder changes, that the folder.jpg does not move along with the media file and the JRSidecar?
It should. I know people have complained about it before...
I store all of my cover art in the file tags, and in a single location on disk. I'd probably rather have the auxiliary image files, when necessary, stored with the source audio files conceptually, but in practice it hasn't really worked very well.
You CAN manage it manually if you work at it a bit. I'm pretty sure Alex has posted a system he uses for this before. Basically, you want to actually import those image files into your media library, tag them with the audio files (so they get the same [Artist] and [Album] tags and whatnot), and then Rename them all at once using a custom view that shows both at the same time.
This is such a common need, though, that it really should handle it invisibly in the background. My guess is that the devs would agree, but that it just hasn't been a high priority for fixing (exacerbated by the fact that there are workarounds that are effective).