headlights not working

The black wire that was burnt up is the same gauge wire as the black wire on the alternator.
That's the alternator output wire. R6 in the diagram.
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It provides all of the power to the car when the alternator is turning fast enough.
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It, and all of the wires connected to the main splice are connected to battery positive. Which means if any of them touch ground the battery will discharge.
If you want to measure continuity with an ohm meter, disconnect the battery.

Volts. Volts are like pressure in a piped water or air system.
A fully charged battery will have just over 12.5 Volts.
The alternator is regulated to have an output 'pressure' around 14 Volts.
Amperes. Amps are flow like gallons per minute in a piped system. The dash gage shows amps in or out of the battery. 40 to 40.

When the headlights are turned on with the engine off, current flows from the battery to the headlight switch. The dash gage should show approximately 10 amps discharge. The battery voltage will drop a little but should still be around 12. Voltage will drop as the current flows through resistance in the circuit.
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At the main splice the current stream splits, some flows to the headlight circuit, some to the parking lights. If you had the door open, some also goes to the dome or courtesy lights. Voltage at the ignition switch, alternator output stud should be the same as at the main splice unless the connection is broken.

While cleaning this mess up, I'd take that halo thing and kick it down the road. The last thing these cars need is additional electric loads, especially on on the headlight circuit. These things were built for 40 W/40 Watt headlamps. For that, the wiring was adaquate - when in good shape.