** Does the circuitry of an igniter actually have a continuous path when not operating? **
Yes. It is a semiconductor, it is essentially a big resister. At a room temperature the resistance of the ignitor should be between 50 and 1100 ohms*.
* NOTE. Just having that resistance does not necessarily mean it is "good", just that it is not open (ie. burnt out).
** I also tested the continuity for the igniter (the one I think I burned out, as described earlier) **
You said previously, "When the "wrong" wiring was in place, there was no power delivered to the igniter because it was attached to itself."
If it was never connected to power it shouldn't be "burnt out" although its element could be physically broken if it was handled roughly.
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