I don't have the service manual copy. I was servicing these sets more than 6 years ago when I was working with a service contractor for warranty repairs.
The degausser is very simple. The AC is switched on with the relay. The relay feeds the degausser in series with the thermisistor. There is not very much involved to trace this out by following the traces on the circuit board.
The common fault is that the thermistor goes bad. You can order this part from any of the parts distributors for Sony parts. I cannot remember the cost of the thermosistor, but I think that from Sony it was in the area of about ten dollars at the time.
It is very rare that the coil or the relay goes bad. On some rare occasions I have seen cold solder connections cause the degausser to not function.
Your set will work with no degausser working. You can order from any place that sells parts for TV service, a manual degaussing coil that plugs in to the AC. You can manualy degausse the CRT face several times per week. This should work out well.
Manual CRT degausser coil:
My preference is original parts. But, if you want to chance an after market part check the link below.
Donberg Electronics sells some types of thermistors:
In the service shop we used a manual degausser when were changing picture tubes. We needed to have the screen degaused very fast so that we can do the purity and convergence set-up.