If the power output is expected to be 1000W, usually you design the system to utilize all of it. So as the power starts to degrade, even when you're really far from the half life, it may not be enough to power the systems.
Specifically, the Voyager's RTGs have degraded a lot since Pu-238's half life is much shorter[0], but most of the machine's systems can't run at ~67% power. We're basically down to the very last few experiments running intermittently.
So for useful energy, parent comment is largely right. You've got <5750 yrs of usable work, which is absolutely fantastic.
[0] https://en.wikipedia.org/wiki/Radioisotope_thermoelectric_ge...
So, don't do that then. Adding capacitance is a problem, dumping heat is not. Seems a small issue (at low power) for a doubling of operational lifespan.
Right out of the gate aluminum electrolytic capacitors are out as getting 10-20 years out of consumer grade ones is pretty good. I have a single monitor I've replaced caps in 3 times now (and they weren't even a known bad batch) in the past 10 years. Electrolytic caps will go bad even faster if they aren't regularly receiving electricity, many have a 2 year shelf life. If you've ever plugged in a vintage electronic (say an Atari/Commodore computer) and you've had smoke-like stuff come out, that's usually the electrolytic capacitors going. If you open up an old device that has had one or more fail, you'll find this brownish-black stuff all over the insides after they go.
Most super-capacitors would die from use long before useful power output, they're usually good for a million cycles or less so even at one cycle a day you're only looking at 2700 years and change.
Polymer electrolytic capacitors are good in use for 10-15 years.
Pretty much every capacitor out there right now has a 5-15 year lifespan.
You actually worked on restoring them? Do public images of the innards exist anywhere from that? I'd love to see that, the innards of old electronics and machines are just so beautiful in their simplicity. You can look at them and actually understand what is going on.
Edit: found your blog, getting lost in the images.
For more information on the restoration, see my blog http://righto.com and videos at http://youtube.com/curiousmarc
But it's primarily wattage - at some point the overall power dips too low to maintain voltage at that amperage to keep the system alive.