One problem is that an armed South Korean national showing up in NK could not only get shot for his trouble, but be used as an excuse for war.
“South Korea just invaded us!”
...and technically, they’d be correct. You can’t let armed South Koreans cross the border into NK.
Of course, at a distance, you might not be able to discern if he was armed or not. Too bad for him...on the DMZ you have to shoot him to be sure.
True, but on the other hand, it’s sort of a propaganda coup for the North : “Look, the South Korean lackeys of the American dogs shoot people who try to leave for the North! Does that sound like freedom to you?”