The cooling/lubricating is internal in the pump so no fuel to lube/cool, no fuel to run, so no pump running after the motor stalls.
If you think about it, if it was such an issue, wouldn't every manufacturer make it so that you always have half a tank of gas when reading empty and a low fuel cutout to prevent a low fuel situation???
Another reason I run mine low, the truck sits for months on end at times without running - so when I do use it (usually for longer runs) I fill up with fresh fuel and run that tank low to get rid of the older gas. Gas w/ethanol ( almost all that is available here) goes bad and attracts water in a fairly short period.
I am not positive, but the fuel in the tank right now might be from late summer.
Not unusual for me to have a tank of summer fuel in the spring and winter fuel at the start of summer.