The tedium of winding coils led me to think more about how to simplify the control circuit. I believe that I can make a modulo 12 counter circuit using 14 relays (6 SP2ST and 8 SPST), and a complete control circuit with 20 relays - I'll post a circuit diagram if it works.

The picture below shows progress so far on making relays:

Some of these are SP2ST relays - the name I give to relays having two single throws which both close onto the same pole nail. Like the SPST relays, these also have redundant paired contacts for each throw:

For places where I need an XOR function, I intend to use the following:

I've been reading a textbook about electrical contacts recently, and learned that the zinc layer on the galvanized iron nails is rather a poor choice to use for contacts - perhaps that explains why the contact resistance I'm seeing (0.5 to 2 ohms) is a lot larger than that found in commercial relays (< 0.1 ohms).