Why does changing scale from 0.01 to 0.1 ruin my animation?

I think this usually happens to the IK bones in your rig. The scale causes the IK bone to experience greater effects.

If you need to scale up your character, I think you should do it in the import setting. It will compensate the import-scale into the animation. I worked with a team of Maya artist for 2 weeks, I think this is how they do it in Unity.

What scale are you changing that isn't the game object?

So I was doing it in Unity's import inspector for the fbx... is this not the way to do it?

I don't know an answer to this, I've never encountered it. I do know that animations contain position information - if you rotate your bones instead of moving them from keyframe to keyframe, the problem should disappear. If they're already purely rotational animations, I have no idea.

I am using skinned animations is this a problem?