@stan2004 posted:That you identified the resistor as being 300 Ohms tells me you probably have a meter? Your photo suggests the resistor is in good shape (not charred/toasted) but a simple meter test will confirm that.
I'd say the failure is because they did NOT install a 1-cent diode to protect the LED from reverse-voltage breakdown from applied AC voltage.
In this recent OGR post regarding a wedge-style plug-and-play LED replacement for an incandescent, they include the diode. I'd contact the manufacturer and refer them to this thread! In bulk the diode will cost them less than a penny.
Separately, and I realize you're running at "only" 10V AC, but a 300 Ohm resistor is suspicious. Your photo shows a 1/2 Watt LED. Let's say it's operated at half of that since it's AC...so 1/4 Watt. The math is a little more involved with AC operation but if operating at its rated 18V and dropping that down to ~3V as per a typical white LED...that means the resistor is absorbing several times the power of the LED and that looks like only a 1/4W resistor.
Yes, I did meter test to confirm the resistor value (which matched the banding). I also suspected the resistor was undersized. The whole bulb casing may well be acting as a heat sink. As you and others have suggested, I think reverse-voltage breakdown is the culprit.
I have a sleeve full of 1.0A diodes. I just crimped one onto the end of the wire that powers the light in my 445 switch tower. It was easy enough and I just trimmed and inserted the end of the diode into the spring clip.
Now for my string of Lionel No. 70 lamp posts: Will a 1.0A diode be good for a string of 3 LED bulbs, or would each lamp require a 1.0A diode? I'm the wrong kind of engineer for making these calculations.