OK, the ones before the 4-pos dipswitch were revs A & B. To nail it down:
At C4, do you have an LS191, or something else?
At F5; LS193 or LS191?
At B7; LS161 or LS191?
In any case, you'll want to get either the 1st or 2nd printing of the manual, and the 1st printing schematics, from here:
http://arcarc.xmission.com/PDF_Arcade_Atari_Kee/Asteroids/
Sheet 2B has the sound circuitry.
Yes, the 555 @ R10 is the one that feeds the saucer sounds. As you'll see on the schem, other components in there include the quad switch @ N10, the hex inverter @ N9, a couple of transistors, and of course the 566.
LM566s aren't very common, nor cheap.
You should be able to eliminate the latching IC and the inverter buy grounding pin 12 of N9. If that makes your sound, everything downstream is OK, and you issue is either N9 or M10. If that doesn't make your sound, the problem lies in the actual sound-generation circuit.