Ok. Looking at the video again, the telltale symptom here is that with the headlights on, the license plate and tail lights go out when the turn signal (amber) comes on. Since the license/tail lights are on the PARK circuit and always supplied with +12v when the headlights are ON, the only way these lights can be going OFF is if +12v is being applied erroneously to the GND side of the license/tail bulbs when the turn signal is ON, which implies there is a missing ground. This +12v might even be sourced through the turn signal bulb(s) if they are not properly grounded, after all an incandescent automotive bulb like a 3157 is like 5 or 6 ohms when turned on.
My best guess, since your center brake light is involved as well: there is a junction block on the rear most cross member, facing the rear, drivers side. The tail/brake/license/reverse/turn signal harnesses all tie in here, there are 4 different connectors. I would dissasemble, inspect, and clean this very well. Look for corrosion or other possible bad connections/shorts. The junction block should be a sealed part I think, if it's been damaged or cracked, moisture could get inside and cause all kinds of havok. Good luck.