I believe the standard production check is more like you check continuity between known nets, given that you're the manufacturer, you already know where they are exposed, and therefore you can perform those checks before adding any components. Post component checks are a lot more complicated because active components and passive components will modify the visible voltages and characteristics, often to the point where you won't have the same degree of physical insight.

I would assume once machines are set up that this is only really done if you're not confident of your manufacturing line for some reason (eg. maintenance, reconfiguration) or you are pushing limits somewhere, for example, particularly small vias or traces very close to the edge of the board.

To make this useful, you would want two flying probes because otherwise it's not going to be telling you much you don't already know.