Views Slideshow Stops Working Alongside Other Slideshows
After some inspired messing around, I determined that the issue only stopped slideshows that were displayed after a certain slideshow. Using that information, I deleted the isolated block, and re-created it in the same form as the other slideshows.

This appears to have solved the problem. I guess something was corrupt with that particular view. It was the one I was originally using as a clone base for all the other slideshow blocks.

Hopefully this "fix" persists!