The signal for that wire comes from one of the small wires on the alternator, without it connected the alternator will never get a 'start charging' signal so will never charge the battery.
At a guess you've gone wrong at the start charge loom somehow... I don't know 13 wiring very well though.
The dash bulb has a positive supply to it, with ignition on it earths through the alternator, when the engine starts and alternator charges the earth path becomes live, the bulb has a live at both terminals and no earth it so goes out. If the alternator fails then the earth returns and brings the bulb back on