First, an Alternator WILL charge a battery. That's because the alternator is a dumb electricity generator. It just maintains a constant level of power to the battery and across the spectrum of your car's voltage/amperage requirements.
But it also has a lot to do with the battery type. Some batteries charge with an alternator better than others.
The old Lead/Acid batteries are the slowest to come back up, and if you just so slightly tip an older battery you've probably stirred up the crap in the bottom and shorted out a cell (or two). A battery of this type is usually considered "dead" at 10.5 volts.
With the "Gel Type" batteries, most are "deep cycle" and can be drained down to about 5 volts before they become difficult to recover, and with the little "Dry Cells", they're pure deep cycle and can be literally drained to nothing and yet recover in about 15 minutes.
It is a very good idea to start out with a fully charged battery (13.2 Volts)
The reason I state, emphatically, that an alternator is a battery charger is because that's what it does . . And it's the first thing it does after you start your car . . You pull about 2.5 volts , on average, when you start your car if it's a relatively new one with a computer, a HEI and a stereo. Those volts have to be replaced, and your alternator does that "on the Fly", while powering up your power seat, headlights, windshield wipers, etc . .
In the case of bixblk, it would sound like a "controls" (computer, ECM, etc) problem, but with todays systems, unless he can chase it down with an OBD box or get lucky and find the right dirty connection (Which is probably 50% of all electrical problems in the modern automobile) I certainly don't have even an educated guess . .