Gradients won't move with their objects, and I have DO have the "Move Gradients Along With the Objects" button turned on.

I've re-started Inkscape and I've re-started my computer, and the problem persists.

Anything else I should try, or is this a bug?

Perhaps: https://inkscape.org/forums/questions/getting-back-to-square-one-and-starting-over-its-easy/

Okay, I think I solved the issue. I'm working with Inkscape on a file that was created in BoxySVG, and something must have translated badly. When I create a new object with the same gradient fresh here in Inkscape, the gradient moves along with the object with no problem.