Before you go replacing any boards, if you’re comfortable rummaging around inside the unit, you may consider unplugging and replugging every board jumper/connector cable. Sometimes it’s a simple as a dirty/weak connection at those plugs. I can’t guarantee it’ll work, but there’s a decent possibility that the signal is just being interrupted because of a faulty connection - especially since you mention that it does work sometimes. This was an issue, easily resolved, in my STR-D1090.
See all those wires running around inside?
View attachment 1215554
Be careful with some of the smaller ones - not exactly fragile, but you don’t want to accidentally rip the wires out of the plugs. Some have locking clips, others don’t. Just unplug, then plug back in - one at a time, and as many as you can get to.
Before tracking down a difficult to find 30 year old part that may or may not solve the problem, you may want to at least give this a shot. My sense with your description of the problem is that you may find it near the balance control. One sure-fire way to rule out balance though, is to put the unit into ‘source direct’ mode - that bypasses all tone and balance circuitry with the exception of the loudness button.
Also, a little more advanced: if you’re sure it’s coming from the volume board, you may consider looking for cracked solder on that board. Reflow it, and you should be back in business.
Welcome to AK, btw.