The action spacein robotics isn't small! That's the misconception of many. Contact makes the dynamics hybrid: you switch manifolds every time a contact makes or breaks, and the optimization landscape is non-smooth precisely there. A low-dimensional command vector is not a small problem. That difficulty is the SE(3) geometry you're waving away!
I'll be giving a "exactly how bitter lesson'ed is all of 3D computer vision?" talk at the Bitter Lessons CVPR workshop tomorrow at 10:30am, Room 3A-3D. Here's a slide with the overall thesis.