The use of a single diode past the switching stage points to a flyback design. If that is indeed the case, full wave rectification is not an applicable concept, and the assumpton that the charge/discharge current in the capacitor has a 50% duty cycle would not be correct except at a specific (and low) input voltage. Flyback designs work well and are quire common, but they do need a decent HF rated output capacitor.
The design process for a switching power supply design does take temps and ripple current into account for the derating of the capacitor. The various switcher controller manufacturers have simulation and design automation software services available to make a design very effective even for a novice.
In my experience most all output filtering capacitor failures are one of two issues. Problem one and the most prevalent is poor capacitor performance due to electrolyte formulation, or counterfeit components. In one issue we had a supplier that attempted to replicate the Panasonic electrolyte with disastrous results. Not one of their capacitors met the lifetime rating listed in the datasheet. The parts were failing at 1/10 of the rating on the sheet. Needless to say this was a big issue costing the company a significant amount of money.
The second issue is that the environment experienced is not what it was designed for. We have had units placed in very high temp ambients, far greater than the posted ratings. A rule of thumb is that every 10C increase in temp results in a reduction in life by 1/2. Given that you bought the unit used, you cannot rule out if it was placed in a high temp environment.
Switching power supply design is a very mature specialty, which can be fraught with pitfalls just like any other activity. While bad designs are out there, sometimes the explanation is much more simple than the derating choices made by the engineer.
Frank, that is an interesting and very good analysis. Your obvious deep knowledge makes it "easy" for you to see the situation. On the other hand, not everyone has such a deep understanding. It might have been useful for the designer to model the system in a CAE tool that can handle multiphysics. The interaction of electrical and thermal effects is very important in electronics, and as you found out, often overlooked.
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