I know nothing about amp fires, so I won't discuss that topic. If you unplug your Rhodes from the electrical outlet after use and if they have their proper fuse, I'd say you're safe.
However, we can talk about repair or maintenance.
Of the electronic components in your Rhodes, electrolytic capacitors have the shortest life span. This type of capacitor has a lifetime of 1000s of hours. This information is usually found in the manufacturer's data sheet. But this varies a lot with actual operating voltage and operating (ambient) temperature.
If you use your Rhodes 6 hours a week, these capacitors will be out of use in 10 to 20 years. If your Rhodes has never been serviced, the capacitors inside are worn out. The situation you describe can be caused by this, but not only...
If you maintain your Rhodes (i.e. change worn components), you will have a more satisfying experience with your instrument.
But what if you wait for a complete breakdown?
In some cases, a capacitor can leak and spread chemicals on the PCB, which is not good for other components. It can also "pop" with the same result, plus an acrid smell. It can cause a short circuit, dragging other components down.
My opinion is to change the capacitors if they are too old. They are located on the 2 amplifier PCBs, on the power supply PCB plus 2 on mounting clamps, and on the preamp. Changing caps is not a big deal if you know how to use a soldering iron and a desoldering pump. You can find maintenance kits on the website of online dealers. If you are not too confident with this, you'd better ask a professional technician because there is a safety hazard with the current here. He could find other issues too, not adressed only by the capacitors.
An example of successful maintenance here: https://ep-forum.com/smf/index.php?topic=10612.0
Keep us in touch.