As the nut does not contain a lubrication "tit", it should be pre-greased by the manufacturer. Leave as-is.
umm, that's fine if something is spinning in one spot, like an axle,
but this is dragging the greased balls up and down a foot of screw thread, I'd suggest that any grease applied at manufacture is going to be wiped off on that rod eventually.
If the balls were in their own isolated compartment, then different story.
Although you do make the point that some ball screw bearings come with a grease nipple, but I suspect that's a cost saving on those that don't, rather than a recommended maintenance idea.
It's not going to crash and burn either way.