I've tried using five different soldering irons for stained glass projects the past three months and they all stop heating up after a short while. The first one stopped working after three weeks and the last one today stopped working after a minute. All of our other appliances continue to work in all of the outlets.

Your using the wrong tool. You should really consider resistance soldering. You'll never go back. With resistance soldering you pass a current through what your trying to heat. It heats instantly. I knew a guy who was in business doing model building using brass. He never went back either.

Simplistic view:

Flux and assemble like usual. You can have a pair of tweezer like tongs and can clamp an object between the tongs. You have a dial that selects the power and a foot pedal. You must make intimate contact to the metal or metal objects to be joined. You then press the foot pedal and pass a current until it heats up. Apply solder. Solder won't stick to the probes. Then hold until it cools. Release tongs.

Soldering Irons, Resistance Soldering, Solder Pots by American Beauty Soldering Tools

I'd also consider the UT-100SiK butane soldering iron as well. There is model without the built-in ignitor. That model will torch, but that isn't necessary.

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But to answer your question. The element of an electric soldering iron can be fragile or the iron could be of poor quality. Constantly heating and cooling can degrade the element. Dropping the iron can degrade it as well.

The bottom line is that it's not suitable for stained glass.