My time is very limited at the moment so I've avoided following your detailed description...
I'll try and see if I can confirm what you're saying re. circuit paths and voltage.
AFAIK, the 'scope is still operational so it should be easy to get an idea about timings. I'd expect them to be several 10's of off if there's a problem iin the integrator. OTOH, good values (few %) would be further support for the sw. pos. 9 circuit failure.
If the timings are sort of ok, even in positions where the voltage isn't understood, I would leave trying to understand that for later.
Still, I'll try and follow your description of the low-voltage findings.