packrat31

Bill, I can't thank you enough for taking the time to map this out for me and share. I have been looking it over again and again to familiarize myself with it. I foresee myself taking on the challenge... with my electrician uncle (I am a novice!)
And thank you, Neil! I'm determined to make this silly Duncan kiln work for me. Having each set of elements on an infinite switch will be a game changer.

This may help and I think this might illustrate Neil’s description above and ensure both legs of the bottom element are disconnected from power when off and sized for the load, it  uses the existing relay in the top Left of your diagram.
Disconnect all the wires with an “x” On power relay “A” move terminal wire #1 to terminal #4 Run a new 10 gauge high temp wire from relay terminal #9 and connect to L1 Run a new 10 gauge high temp wire from element connection to relay terminal #6 Install new infinite switch and run h2 & h1 to power relay A “coil”  this will cycle power relay A and in turn the lower elements per the setting on the new infinite switch Intercept L1 & L2 as shown and connect to new infinite switch. Note since these are normally closed, relay B is just fine as is unpowered except for the coil connections.which should be removed. Note all timers and high, medium, low switch will be entirely disconnected and non functional. Pretty sure that  all works, maybe folks here can double check me.

Assuming the wiring diagram above is the correct one for your kiln, you've got 3 sets of elements and only two infinite switches, so you'll need to add one more switch.  Just drill into the box and put it wherever it fits. The top section and middle section can connect directly to an infinite switch. The bottom section cannot connect directly to a switch because infinite switches max out at 15 amps and the bottom section pulls 22amps. So for the bottom section you'll have to use a relay as a go-between. The infinite switch will cycle the relay on and off. You may be be able to re-use one of the relays currently in the kiln.

What position is your 3-position rocker switch in?

Unfortunately the bottom elements are hooked to a normally closed relay controlled by the repeat cycle timer. Since the relay is normally closed, the repeat cycle timer actually opens this switch turning off the elements. If the repeat cycle timer does not cycle about once per minute, then the elements stay on. I couldn’t agree more with Neil, adding a third infinite switch is probably the best fix I have seen for these. The timers are expensive and mostly extinct and really did not provide clarity in operation nor any real automation to this kiln.

Welcome to the overly-complicated world of Duncan Kilns! I can't get you any definitive answer on how to diagnose the timers on these old kilns. IMO, the best option is to do away with the timers completely and convert it to a standard manual kiln with either 3-way or infinite switches. Some of these timers are still available, but apparently some of them are not. There are a number of folks here on the forum who tinker with these kiln, so someone should be able to help.

I’m commenting to help boost this.
I’m not the biggest expert, but the timer on a manual kiln should override the kiln sitter and shut everything off. In fact, your kiln shouldn’t be able to turn on at all if the timer’s at 0, even though you can carefully override the sitter. 
If you need more technical info on your old Duncan kiln, Paragon took over the servicing of them years ago, and they do have all the manuals and wiring diagrams for your model. In the operator’s manual I found there, they do have all the troubleshooting steps to diagnose where a timer switch or a relay might be going sideways. https://eadn-wc04-7751283.nxedge.io/wp-content/uploads/LX_914_Duncan_Kiln_Owners_Manual.pdf