Still voting for the spark gap. See if you can find any numbers.
https://www.newark.com/search?st=spark gap As the gap erodes, the breakdown voltage gets higher and the circuit fails to start. There is an application circuit in the datasheet,
The spark gap should briefly light up. The voltage across the capacitor could tell you an upper limit. In other words, it has to be less than that. Measuring the voltage across the capacitor is VERY DANGEROUS. and could be
FATAL. The capacitor is LIKELY the rectangular thing near the bottom of the daughter PCB. Expect something near 400-800 V so test probe insulation breakdown is a REAL ISSUE.
It would not be an issue for me since I was used to fixing 100 kV, 0.1 A and 15 kV at 1.5 A DC power supplies and 1000 W RF tube transmitters with 3000-4000 V plate voltages professionally. In high school, I built a 3000 V power supply for a science fair project and I've been working around picture tubes since I was a kid.
The cap should charge by a relativey high frequency oscillator and then break down by the spark gap. The fast pulse is applied to the primary of a transformer where the secondary is in series with the electrode. The transformer has a high turns ratio causing a very high voltage on the secondary, probably 10's of kilovolts.
The one hand in the pocket rule always applies.
I'd try to make the measurement when not actively powered and shortly after an ignition attempt with power off.
The high voltage pulse could damage your meter. One lead you can likely connect permanently if the insulation of your test probes is good enough. The second connection would be made using a long piece of PVC pipe holding the test probe.
i.e. Make a fixture, that would allow you to probe the voltage across the cap quickly after an attempted ignition and th the power off.
Normally, i'd make connections with the power off, but not in this case.
Your cardboard thing is a transformer. Since it's basically in series with the electrodes, it has a high turns ratio, but the gauge of the wires are vastly different.