Why do some Canon lenses stay wide open off-camera while many Nikon lenses stop down when removed?

Both systems (and all others that set the aperture via the camera body) will set the aperture wide open when a lens is attached to the camera, to provide the brightest view possible in the viewfinder and to facilitate manual focusing.

Canon EOS lenses uses an electronic iris. A signal is sent to the lens to close the iris when taking a photo or using DOF preview. Since the aperture is wide open most of the time, the camera or lens would have to detect removal of the lens and send a signal to the aperture control motor, which is extra work for no benefit.

Nikkor lenses use a mechanical coupling which is spring loaded. Removing the lens means the camera body coupling is no longer pushing the aperture control lever, meaning it springs back, closing the aperture. Designing the system to avoid this would probably have been additional work, again for no real reason.

In summary any slight advantage to having the aperture open or closed when the lens is not mounted is dwarfed by he design effort involved due to the different approach of mechanical vs electronic control, so the manufacturers did what was easiest.