In most cases, the problem with an undercharged battery or a "weak" starter, even on a new battery, is caused by the faulty voltage regulator - it is located in the alternator and is replaceable; google: audi-volt-reg-06h903803e. No need to replace the entire alternator.
In my case, something else caused the problem; the "primary hot wire" (p/n: 8K0971228) connecting the alternator, starter motor and battery. Its central, "pressed" connector degrades over time - it is made of aluminum; google part number with google images.
In the video above - I used a bypass of the degraded connection using a new 25mm^2 cable.