The critical angles that a lapidary uses to facet the gem are based on an air/gem interface, not a metal/gem interface. Setting the stone into a metal interface with the gem surfaces will reduce the brilliance of the light being back our the table due to the internal reflection/refraction of the rays. Thus, not drilling through the metal will probably reduce the beauty of the stone.
Well, yes, but…
With at least higher Refractive index stones like diamond and a
number of others, the total internal reflection from facet surfaces
that gives stones their brilliance depends, as you say, on the
air/gem interface. Not drilling the metal under the stone does not
change this, and if the stone is clean, you cannot then see the metal
under the stone. Not drilling the metal, if the stone is well sealed
by the flush setting or bezel or pave seat means less dirt can get
to the stone, so it stays clean longer. But dirt and grease being
what they are, eventually they’ll get in there, and the pavilion of
the stone gets dirty, no matter what, and then the beauty of the
stone suffers. But it’s mostly the dirt, not the metal under the
stone, that does this, at least with stones having a high enough R.
I. to totally reflect light back out the table. With lower R. I.
stones, where there is some “read through”, meaning you can see
what’s behind the stone even when the stone is clean, then of course
whatever is behind the stone affects the beauty (thus the practice of
“foiling” gems back in times before modern optics were understood in
facetting. You still occasionally see it with tansparent cabochon
materials…, or similar practices such as darkening/oxidizing the
metal behind, for example, opals, to increase contrast) Then, what
the metal behind the stone looks like becomes a design choice, to
include knowledge of what it will look like when dirty. The biggest
difference, I think, is simply that we pretty much know that no
matter what, the pavillions will get dirty, and brilliance will
suffer. Drilling behind the stones means it’s then easy to clean the
jewelry again, restoring brilliance. With lower R. I stones, drilling
or azuring the metal underneath means you see open space, not
reflective metal, under the stone. Often, this is a better look, but
not always.