Idiot light circuits like the one you have typically have one bulb and one resistor in parallel to it supplying exciting current to the alternator in order for it's field to generate enough magnetic flux to generate it's own field flux. This results in the light going out since it is no longer supplying enough current to light the bulb.
If your resistor burnt out or was never hooked up you would have exactly the situation you describe, and the fix is certainly a simple one. Just add another bulb in parallel to the one you already have and the additional current supplied by it will cause your alternator to start charging as soon as the engine starts. I prefer to use sealed trailer clearance lights because they are just the right size and can be had for cheap. You can just wire it up to the two wires that are going to your current idiot light and leave it under the dash.
Whenever I add an alternator to a system that didn't have one before I use a standard Delco SI-12 55 amp with two clearance lamps (wired parallel) between the ignition switch and terminal 1, terminal 2 gets wired to the output directly so it's hot all the time. Your Bosch complicates that picture a little but not from the standpoint of the existing idiot light, just double up on it and you've done the same thing.
It's really that easy. The parts store will have splicing couplers that allow for easy hooking up to your existing idiot light wires and the added trailer light socket with bulb. If it don't work you can cut it off easy enough, either way a half hour with the right parts and you are done.
It is NOT good to have to rev an engine to get an alternator to start charging, it creates an incredible voltage and current spike to match that will over time take out the voltage regulator and cause the alternator to charge at full output until it burns out itself. One bulb systems do this constantly and after a couple years of spikes, I had to put in another bulb, and regulator, I've been a two bulber ever since. No more problems with spikes or pegged ammeters going taaawaannggg!