As observed from your picture, the white/black wire seems connected properly. Since this wire travels from the switch to the coil (and then to the distributor), a break along the way may be intermittently connecting to ground as a result of vibration. I would suggest you disconnect this wire at the coil to test this theory.
As far as the cutoff switch is concerned, this switch, as most of the Healey’s electrical components, can be disassembled, cleaned, lubricated with dielectric grease, and reassembled as good as new. Since your car runs without the switch in place, it appears your switch’s internal connections may be faulty and your switch may need to be cleaned and lubricated or replace if the contacts are beyond cleaning. Corrosion is a common problem.
I would also suggest you do a number of Voltage Drop tests to validate all your major connections are in proper condition. Voltage drop tests will show the condition of your wiring and connectors by indicating how much power you are loosing when passing from one point to another. I recommend running a set of these tests to establish a baseline before making changes. If you need it, there are a number of Voltage Drop DIYs on the web for you to choose from.
Good luck,
Ray (64BJ8P1)