Joel:  I can't say for sure what is causing the zinc disc to play
poorly without seeing it play.  However, because zinc is a softer metal
than steel, it may be that the projections have been depressed just
enough to cause this condition.

That is, they may, as you suggested, "not be deep enough" (not project
as far as the ones on your steel discs).  Try measuring them to compare
and let us know.  As you may know, zinc was used by many disc box
manufacturers early on, but replaced with steel because of its poor
quality; projections were regularly popping off and customers were
likely complaining.

Now to your question about having the case of such your Regina music
box professionally restored.

My personal opinion is that if it sounds good, leave it alone.
However, I have been known to refinish a case or two.  But when I say
refinish, I don't mean "make it look brand new".  I have seen a number
of beautifully refinished musical boxes, but they look like they were
made yesterday -- they no longer look like antiques.

I prefer to refinish them by removing the varnish, usually alligatored
or built up with years of dirt and grime.  This can be done using fine
steel wool soaked in alcohol.  Keep the steel wool clean by constantly
dipping it in clean alcohol.  This can be a tedious process, but it is
a necessary one.

Leave the original finish alone; that is, do not sand down to the wood
or even remove the original stain.  Rub in (sparingly) some matching
stain to try to cover deep scratches that go down below the original
stain, like the cat scratches you mention.  Then apply varnish and wax
(or French polish).  This way you leave in the patina, the "age marks"
like bumps and rubs, which give the piece its antique look.

But most importantly, whatever you decide, enjoy the music!

Best Regards,

Bill Wineburgh
http://members.aol.com/WWineburgh/musicbox.html