First off, have the headlights ever correctly operated during your ownership or is this a recent equipment failure?
Your troubleshooting steps have covered just about every base short of wiring harness continuity. 👍👍
when you say you have checked the Continuity switches, that means you’ve checked them to confirm they can attain both open and closed positions, right ?
studying your photos I think you will correct your problem by connecting the red/black wire to the terminal between the red and brown wires. The red/black wire connects to the relay terminals, the smaller red wire connects to the relay coil.
The twinned black wires require you to look through the cover of your relay to see if there is an internal jumper from these two terminals.
If there is a jumper then you are good to go. If not, you need to add that jumper or split the two blacks and connect one to each terminal.
One additional suggestion. If you have the jumper, it is better to connect the two black wires to the terminal next to the gray wire. That is the terminal that connects to the relay points. If you connect to the terminal as shown in your photo, with an internal jumper, you are running the headlight motor power through the small internal jumper - hi amperage with small wire should be avoided.
Larry