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Your videos are exactly what my lights looked like on the chinese-crap switches. I bet it doesn't do it with just the low beams does it? My low beams worked fine but as soon at I hit the high beams, more current and the thermal switch couldn't handle it.
Basically what you are doing is taking the output wires from the switch and wiring them to the coil side of the relay. This in turn takes battery voltage on the other side of the relay and send it to the headlights.
Also a side benefit is removing that large current load out of the switch which to me is a fire hazard.
Relay basics:
Here is where I put my relays and self resetting circuit breakers (gold boxes).