I have tested the tube and starter with a conventional magnetic ballast and they work fine. The tube starts and lights normally so I think it is worth continuing with the repair.
I have disconnected the zapper spiral wires in case it might be a short in that part but no, it makes no difference, it still does not work.
I have measured, in-circuit, the capacitance of the four big capacitors, the three yellow ones and the brown one, and they are all within specs.
I suppose I could test the rest of the components and that would be the quickest.
Or I could reverse engineer the schematic and learn more.
Edit to add: After a cursory check all components seem to be within specs. The two green caps which I am assuming are part of the charge pump are marked 2J473J which I am assuming is 47 nF and they measure about 49 nF. Resistors, diodes, etc all seem to be OK. I guess it is time to reverse engineer.
Another idea: When testing the starter does glow but the tube does not start. The current is flowing through the filaments but the voltage between electrodes is not enough to start. With a regular magnetic ballast it starts but in the zapper circuit it does not start. It seems voltage is insufficient. I wonder if this tube is at the extreme of voltage needs and would work if I feed the device higher voltage .... hold my beer just a moment....