Question Animated .fbx: from Unity to Blender, and then back to Unity without it breaking?

I have a rigged and animated model in Unity. I wish to make some modifications to it which can only be done in a 3D editor, such as Blender. But the very act of importing into Blender, then exporting back into Unity messes the rigged model up in such a way that the old animations don't work properly anymore.

I feel like I've made one successful step in solving this, which is to get rid of the incredibly annoying "armature" parent that Blender gives every .fbx skeleton on export. I.e. if it was "Root>Pelvis>..." to begin with, when exported out of Blender it's now "Armature>Root>Pelvis>...". I fixed that - with the help of Alessandro Ribeiro's Blender fbx addon. But unfortunately that alone wasn't enough to solve the problem.

Has anyone successfuly and ultimately solved this seemingly simple problem, which I'm sure is very ubiquitous?

Just in case, here's a unitypackage of the model in question. It also contains an Idle animation to test with. If imported into Unity, this model will be working fine with the animation. But if you were to import the model to Blender and then back to Unity, the model is weirded out by the animation.