The individual spotlights have 'Chip On Board' LED's which are not normally intended to be user serviceable.
I see the LED's are individually 12V 3W which equates to 0.25A per light.
You say you've checked the voltage - was that there at the end of the chain of lamps when they are all connected? If so, some of my conjecture below may not apply/ or be wrong?
I would be inclined to disconnect the chain of lamps, and just connect each one individually to the supply on its own, and see if any come to life. If none do, a bit of further investigation to establish if the supply is giving 12V with a lamp connected, and what the current taken by each lamp is. That would probably require making up a test jig.
An option might be to try 12volts worth of batteries across each lamp, with polarity the right way round!
You may have a faulty power supply that gives 12V open circuit, but not on load.
Or you may have one or more lamps failed, or taking excessive current causing the power supply to shut down when they are all connected. If they've all been flashing in unison, that's symptomatic of the power supply experiencing a problem, either overload, or internal failure. But that doesn't tally with your observation there's power to each light. is that 12V?
Or you may have water in the cable somewhere causing a similar effect to above, although at 12V less likely!
Your choice is either some methodical detective work, or buy a new set (possibly a different brand!)