The rail mounting screws seems to be 3/16-28.   The head is 5/16".
They're 1" screws.

The only thing stopping the drawer is the end of the dado where it
hits the drawer faceplate.  The dado in the drawer not only stops at
the faceplate, but also gets shallower toward the front of the drawer,
causing it to jam a bit.  I don't know whether that part was
intentional.

I also have never had a drawer pull, which might have gone in that
oddly off-center hole.  But there's no evidence in the way of rings
or circles.  And the inside of that hole in the faceplate is splintered,
much like the other mystery holes in the sides of the drawer, making
me think it was drilled later.