Bear with me as I lead into my point by way of example.
Over the years with Bosch Alternators I have noticed that they can fail to charge (worn out brushes on the voltage regulator) without illuminating the Light in the cockpit. So in my 1981 Volvo I installed a Voltmeter gauge. After start, the needle always went to 13-14 volts (where it should be for a charging condition).
At 175k miles, the factory Bosch alternator failed. As this was a daily rusty driver, I purchased a Duralast alternator for my Volvo. They are "cheap". And that is probably the operative word.
Mine exhibits the same behavior your does. After start it shows about 12-12.5 volts (no charging). If I pop the throttle above 2000 rpm the voltage gauge shows that it is nwo charging. You may want to try the same thing. However, since I don't know how many rpm the alterantor is at with 2000 rpm at the crank, your rpm may be higher or lower to get it to charge.
I'm going to make a guess here. The Duralast design probably had cost as the top priority. That includes the regulator. The regulator provides an excitation function that gets the juice flowing. Mine requires more rpm to get going than does the OEM speced Bosch. I've test my theory on my three other Basch based systems. After start, with no increase in rpm, there is charging voltage.
If the battery isn't being charged, then your running off the battery and it can not self charge just sitting there.
Please let us know if my experiences are yours too.
S, J.R. Brown
2000 G500 LWB Obsedian Black
2005 Toyota Tacoma Access Cab Off Road Sport
1993 Volvo 240 Sedan Anthracite
1980 450SEL Champange (owned it for 15 years. Great car)
1986 280GE LWB Anthracite (Sold it and kinda wish I hadn't)