The battery appears to be weak but ok for testing purposes only.
The ignition switch had no electrical damage but did have some corrosion on the contacts from water, which was easily cleaned with sanding, then re coating with dielectric silicone grease.
I inserted a 2Amp short circuit finder into the ignition battery feed circuit to test some components before replacing the burned wires. The two amp is low enough enough power to not harm anything, but high enough for dynamic circuit testing.
A note about ground wire circuit color; black is used in the USA, brown in Europe, green in China-- these are sold colors w/o tracers.
Checking the chassis wiring diagram shows that the main power circuit is 20amps (D 5.5 in diagram) and the switched is 15 amps. The location of the main power fuse is drawn to be in the starter solenoid (relay), but this is false.
Jumpered the feed and switch power circuits. Depressed brake lever & checked power to starter solenoid using a circuit probe- No Power. Depressed other brake, - no power. Checked power @ dash wiring connector, both circuits - ok. Checked power into micro switches - ok. Power out - none, switches bad.
Checking starter solenoid, but first remove the plug and add a little oil to cylinder to lube rings, it has been sitting too long. Just good precaution.
Disconnected one brake micro switch, jumpered, checked power at starter solenoid - ok. Momentarily depress start button, starter engaged.
Checked fuse block (4 fuses) 3 ok. 4th is AC powered for lights from ac generator (stator). circuit ok. Ac generator - unknown.
Testing for ignition primary: tested power and grounds to CDI and ignition coil. All ok.
Here is a little tip when checking power and grounds with a circuit probe. Normally you would check power by grounding the circuit probe and probing a power wire. But to dynamically test ground circuits, connect the probe to a power feed (battery +) and then probe the grounds.
Bypassed solenoid circuit to crank engine testing spark. No spark. Tested AC power from crank sensor with DMM- weak output. Tested coil primary & secondary windings- no shorts-ok.
Cranking speed is very slow, even with the plug out and this could influence the crank sensor output.
Next on the agenda, get approval for replacing the burned wires, adding an ATC 20 amp inline fuse into power circuit, and a replacement battery.
Here are some things to help others on the forum:
Wiring diagram- this is one of the best I have seen and it is extremely close to all GY6 scooters (50cc & 150cc) as well.
Photo above of an acceptable DMM & circuit probe for the casual user.