I need to check an assumption.
If the following is true:
The alternator is a working one, and the big fat wire from the alternator main terminal to the battery + terminal is good, and the wire from the alternator D+ terminal to the GEN idiot lamp is good, and the connection from the (take deep breath) GEN lamp through the ignition switch and then back to the battery + terminal is good, and the ground connection from the alternator case to the battery is good (battery and transmission ground straps both good).
And: The engine is running at over 2000 rpm, so that the alternator is charging the battery just fine.
The GEN lamp should be dark. This is because the voltage at the alternator D+ terminal and the battery + terminal is the same.
Then you cut one of the ground straps. The alternator can no longer charge the battery and it starts to discharge. But (is this correct?) the voltage at the D+ terminal and the main output terminal of the alternator are still at the same voltage . . . and since the GEN lamp is connected between these two terminals, the lamp should stay dark -- yes? no?
In other words, will a failed battery ground strap on the battery, or the transmission, light the GEN lamp?