My puppet's Position X/Y read 0 in Transform, with no keyframes and no Parameter Recording enabled on either. I can manually drag the puppet into the correct spot in the canvas, but the instant I move the Timeline pl

Dragging is not the ideal method to position the puppet on the stage, as Character Animator has kind of reserved dragging input for positioning of the puppet’s individual features. To get the puppet where you need it to be, you will need to modify the x and y values until the puppet appears where you need it to be.

Another option would be to add a camera to the scene and then position the camera so that the puppet appears where you want it. This link has more information on the Character Animator camera: Control puppets and background together | Character Animator