[NoDOHC]

Actually, you need that warning light wire. The alternator gets the initial current to excite the fields from that wire (through the light). Without it, you are relying on the residual magnetism of the alternator to initiate excitation (may not be sufficient, especially because the control transistors are connected in a Darlington configuration).

I just added length to the factory wiring and connected the appropriate wires to the alternator.

It might work as you proposed (if the residual magnetism is high enough), but I would advise connecting both wires.