it is an unusual setup to the volume control, that the wires are not
directly soldered to it, but transferred through those 2 springy touch stems
sort of like they used on train sets ! the volume control also relies upon
the aluminum grill being well grounded (that little copper wire and clip)
transferring ground through to the Potentiometer body
to test, i suggest you simply alligator clip a jumper wire to the 2 white wires
that connect to the springy stems.. this will bypass the volume control and should
equal FULL volume to the jack for testing.. the grill can be left off for this test
if you still have no sound, the old Mic element may be shot, OR somewhere
along the wire the (white stuff) insulation has cracked off and shorted to
the braid (which is ground, yes) a very common problem after 60 years
the Mic is inside the bellows, you need to remove the bellows pins and open
it up to see it/work on it
it is a pretty old LoDuca since it has round aluminum action arms, a 3 reed
LMH student model looks like.. probably still fun to play.. if you get the mic
working for the heck of it, cool, but that technology was just basic so no loss
really if it's dead
good luck