I don't think this problem has anything to do with that speedo wiring by looking at the diagram.
Ok,
Take a look at the picture below.
The red arrows point to the brown wire which powers the tailight running bulbs.
The blue circle shows where there's a factory splice, at that point the color changes from solid brown to brown/striped.
The brown going the other direction is coming from the park light fuse, that works.
IF this wire diagram is correct this means the only possible cause for your not working with key on is a bad connection at that splice, leading to the ignition switch, the switch itself, or power TO the switch.
Checking that brown/striped for power will help narrow down the problem.
Could be no power on the brown/striped FEEDING the tailight BEFORE the switch also.