You aren't missing anything, the reason the clutch will not come on is either the wire or the coil can not flow enough current to attract the clutch plate to engage it. One strand of the multistrand wire can give you enough current to have your meter read 12V, but cannot flow enough current to energize the clutch coil enough to pull in the clutch plate. You could also have shorted turns in the coil which could increase the current slightly, but not enough to pull in the clutch.

If the above fails, give the single pin connector on both sides a tug apart to see if the wire slips out of its crimp. I had one bad crimp that made contact but dropped the voltage from 12.9 to 9.7 V.

You could use an ice cube relay and fused line from the battery plus to provide the coil voltage and use the coil voltage wire to energize the relay. This is a good solution when the vehicle wiring has unneeded current drops. This will also help the clutch last longer as it assures the full pull into the driven plate.