[zosoppp]

I believe that there is an auto-resetting circuit breaker internal to the head light switches.
If that's the case and the breaker is either defective or is breaking for a reason it might turn the lights out when the switch is on with the lights, but might work when the lights are not connected (when you are checking continuity.

Perhaps unplug side's headlight connector and see if the new switch will run one of the lights. If it will but not run both, I'd think it was a defective switch.

Of course if you've already done a headlight relay modification then this likely won't apply.