Not picturing how the counterbore helps except for hiding a countersink that would "show" when the bolt is not indexed. If counterbored at an angle not perpendicular to the tang, won't the head still need to be filed at an angle that will be revealed when not perfectly indexed?
It helps, but doesn't fix the problem
As Jim mentioned, as the gun wood shrinks, the screw gets tightened, and goes out of timing. If it's a flat head screw, you now have a sharp edge revealed, and it really stands out.
But with the fillister head, you can dome the top of the screw, and it won't expose any sharp edges, no matter how far in it's turned.
However, if the angle of the screw to the flat of the tang is extreme, you may need to shape the screw head to the tang. Then it will show when turned beyond its sweet spot. That said, it is a little more forgiving than a flathead screw.
Tom