If barrels do not "ring" like the bells of St. Mary's, does this mean they are unsafe to shoot, or is this more of a cosmetics issue?

The tubes may or not be safe. The ringing indicates whether or not the ribs are loose at some point along the barrels.

So, barring some other defect in the tube themselves,they should be safe to shoot, even if they don't ring well?

I would advise against shooting with loose ribs. If the tubes aren't damaged by corrosion, dents, etc then they should be okay but the more you shoot it the more the ribs could loosen or the forend hanger which would be a mess. I had a 16ga and a 12ga 00 I shot alot with loose ribs and I ended up having to relay. The barrels are held together by soldered spacers which should keep them together however they could be compromised if corrosion from under the ribs have loosened the spacers as well. At the muzzle end the barrels are held together with strength coming from the ribs, keels, and solder. I relay my own but if you go ahead with the repair it's not the cheapest thing to have done. Zachary

I use a 6" wooden dowel to lightly tap the barrels. Occasionally the extractors need to be removed to avoid a dull rattle.
If when tapping the full length of the barrels there is a distinctive and localized dull sound, that does suggest a loose rib at that spot.

For interest, I just tapped a barrel set with a burst mid-barrel and they both rang like St. Mary's!

Using a brass hammer

Many times a set of extractors or ejectors, which move about freely in a set of barrels when they are not attached to the gun, will dampen "barrel ringing". If your barrels don't ring when struck lightly with a non-marring item while being suspended from the hook, try removing the screw that capturers the ejectors/extractors, pull out the ejectors/extractors, and then try ringing them again. You might be pleasantly surprised.