What advantage does clicking a menu separator have?
History.

In the original Macintosh's System Software in 1984, Apple decided that releasing the mouse button over a menu separator should dismiss the menu. This was probably because they viewed holding down the mouse button as a form of concentration; letting go of the button meant it should go away. Where the mouse was didn't matter. You could even be outside the menus entirely, but not pointing at anything particular on the screen.

Later, when Apple made menus click instead of hold, the behaviour was copied over. That was years ago, the the behaviour is still the same.

(It actually probably predates 1984; the Lisa probably did the same.)

I think this was absolutely the right behaviour for click-and-hold menus, but click-and-hold requires more coordination than click.