Removable self-stick overlays are now a thing. These are printed on clear plastic and need to be cut out.
Instead of cutting the backing with the overlay, the recommended approach is to start peeling off the plastic and cutting each overlay and re-applying to the uncut backing sheet.
Because of the clear plastic, there is no need to cut exactly along the overlay edges, so cutting is faster and easier.
The photo below (enhanced for viewing) shows how the cut overlays can be kept on the backing sheet.
These sticker overlays do not mar the surface of a mapboard, even after several days. Because the centers are clear, you can quickly align the overlays to the centers printed on the board.