Lets try this agian. This time with a little more reading on my part. I reread and watched the video paying attention this time.
What I noticed is the following. First whne you turned the key on the temp and fuellight came on and timed out. What I did not see was the netural indacator did not stay lite. SO either its not in netural with the key on or theres an issue.
What year do you have as it make a diff. You stated two blue boxes so I will assume an 89
Lets start here. 19-2 collant/fuel waring system (1) unplug and see if the lights stay off if not theres a short.
Next make sure its in netural and look for the light. If yo have it in netural and the key switch in on and there no light than refer to 19-6 and test the switches at the plug.
If an 89 the power that feeds the instrument cluster also feeds the timmer unit and is the fuse on (R) wire. since the power that supplies the timmer unit is lighting the cluster via the timmer I will assume you have power to the netural indacator light.
If you look at 19-0 for an 89 it showes the one line for the system.
Next move to 18-0 Electric starter.
looking at the diagram for the 89 you will see that (R) feeds the instrument cluster. At this time see if you reverse light works with the key on. If yes and the netrual does not look at the wire on the switch or the switch its self. You can unhook the netrual switch wire and touch it to ground to see if the light comes on. Note here the ground on that switch also activates ( grounds)the the starter relay.
If the pilot runs via pull starter then the wiring in question is whats on 18-0 diagram.
Since you heard a click and then nothing the blocking diodes are doing there job and preventing a back feed from a loose ground on that part of the circuit.
If you go to 20-1 which is the 89 wireing diagram than you can go from the fuse on the (R) wire and follow it to the ignition switch or bat 2 than look down at the bottom of the page and look at the ignition switch and you will see (R) makes to DC or the (BL) wire which feed the light switch, fan,timmer unit and instrument cluster lights. If your head lights work you have power, If you fan runs you have power through the circuit. If not than the voltage is dropping out through teh circuit and tack it down following the wireing diagram. The plug connections are shown as a block.
This only for the 89 if 90 than I will revise as the wireing harness and protection is different.
If you get a netrual light and no start than look at 18-0, 89 wireing diagram and trace out fuse (RW) after ignition switch and run/stop switch . I say after due to the
Engine runs and the (RW) feeds the
CDI (Capacitor Discharge Ignition) and if you have no power there it will not run.
From the ignition switch it goes to the run stop switch and then to the starter buttonon the wheel from there it goes to the starter relay.
Hope this helps this time and again sorry for the piss poor first post. If this does not work P.M. me and I'll send you my number
If this is a 90 letme know and I will repost.