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This is a low voltage transformer that feeds the lights. Unplug all the light strings that are currently plugged in. You most likely have voltage to outlets but a shorted light string can pull the low voltage power down to nothing so the other light strings won't light. Pull all, then plug in one at a time until one shuts the rest down. That's the bad one.
Low voltage systems can 'pull down' under shorted conditions like you described.
A short on low voltage can go to zero volts when shorted, then come back up to 12 volts when short removed from system
I tried that...I went ahead and cracked the case and discovered that the transformer itself was smoked.