OK, after much button mashing, I figured it out:
If you click on your main FBX object, in the inspector you will find 3 buttons at the top. The ones we care about are Rig and Animations. If you go to Rig and drop down the "Root node" selector, then choose your root bone, it will open up some more options under the Animations button. Go back to the Animations button and pick one of your animations. You will now see a series of Root Transform (Rotation / Position) areas, just under Loop Time. Check Loop time, Uncheck Loop Pose, make sure all the "Bake" options are unchecked, and change all the Root Transform "Based Upon" dropdowns to Original. This is apparently what was screwing me up... some were set to "Root Node Rotation" (seemingly at random), and this was causing me to loop in place or move opposite the root bone direction.
I'll post a new vid here once I make it, showing the correct procedure from start to finish for those searching for a similar issue. Thanks!