I move mine out of the turner, not just turn it off, when it is time to lockdown. I know that some people hatch eggs in cartons and some on mats/screens. I think that hatching in a turner would likely make a mess that is tough to clean.
The difference in success may be minimal, but I also set my eggs so that the deepest dip in the air cell (when held fat end up) is at the top when the egg is lying on its side.