If you find that you have power on the Black with white stripe wire (positive battery cable) at the starter solenoid, then I think that there may be a problem with the White with black stripe wire that feeds power to both trim switches.
That wire is one of the wires you took off of the stud. It supplies power to both trim switches AND to the ignition switch. So, if it parted or went to ground, it would stop the starting and running of the engine.
You said all fuses are good but I would go back and check the #5
(10 amp) fuse for any signs of damage.
Based on your hearing arcing when the outboard was trimming down, I suspect that where that wire goes out the outboard and up to the console is where it may have failed.
Check very carefully where the wiring from the remote control enters the outboard for any signs of chafing or cracking.
That wiring harness goes through a grommet where it enters and the issue might be hidden from view due to the grommet or the insulation of the harness.
Keep in mind though that, when an outboard trims up and down and, in addition, turns left and right, there are MANY places that the wiring harness will flex.
Look with that knowledge for signs of deterioration in any place you suspect that could be happening. It's surprising where a "stress point" will be sometimes.
Good luck.