To summarize:
The pre-amps are working in both channels as verified by connecting them to a different final amp (post 20);
If you jumper the speaker leads at the isolation relay both sets of speakers work (post 23). This effectively eliminates any problems with the speaker selector switch downstream;
Does the relay itself ever operate (close)?
What is the condition of the 10Ω/2W resistors on that board?
When you jumper the two traces together both of the speakers are being driven by ONE of the final amplifiers. The other (dead) channel is also being loaded onto the good channel, so with the lower speaker impedance and extra load it will almost certainly get hot. Avoid doing this or you will have a completely dead unit.
If you check upstream of the relay toward the power amplifier (X07-1450-10) you may find that the left channel final amp has some problems. Check for good audio signal on pins 9 and 10 on that board. Those are the left channel audio out, just as pins 19 and 20 are the right channel out.