You always need some small container on the bench to hold the screws you just took out. Mine are often plastic jar caps, but they’re light, slide around, and let screws escape (often to the floor).
The first serious fix was to glue a large washer to the bottom for weight, and then a layer of non-adhesive but very sticky rubber on the washer. That accomplished the goal – it never slid around at all.
But getting screws out was a hassle, with the threads trapping the screws. An ideal tray would have a smooth cross section. But how to make that?
Pretty good, but smoother – and glossy! – would be better. So some gloss spray paint, and I had this!