Check the 5V power supply. If it's broken, or if it sags, then solo is not going to happen. Any solo switch in the consoles does the same thing: turn on an LED, dump the audio to the solo L/R busses, and dump +5VDC from this supply onto the "Solo DC" buss, which is a control line that trips the lone analog switch (IC9, probably a CD4053 or similar) in the Remix master module. If you don't have stable 5V getting to the chip, then individual solo LEDs may still turn on, but the analog switch won't activate. Occasionally I've had to take the frame apart and solder a broken jumper wire which carries the solo DC buss or some other buss from one backplane PCB to another. Bad solder joints and intermittent contacts are not unheard of in these flimsy frames. If not supply or busses, then these switch chips themselves fail often enough to merit replacing. Also, sometimes a short in one of the channels somewhere is pulling up/down the 5V supply or solo DC buss. Check DC levels and then use your ohmmeter to test continuity from one end to the other. You may have to empty the frame and then reinstall one channel at a time to find the culprit.
Re: power supplies, these consoles do sound better and perform more reliably with a power supply upgrade (I recommend Acopian Gold Box A Series), but you also have to upgrade the power supply distribution to achieve the full benefit. This lowers crosstalk and noise as well. I recommend Coral Sound for a full service Trident rebuild and upgrade package.