New Alternator? battery keeps going dead? So I think you are saying you can charge the battery with a battery charger and start the car with that battery. AND, you measured less that 13 volts at the batery with the motor running.
If so the battery is not likely the problem. Start the motor. Measure the voltage at the battery. If the charging is good you will see over 13 volts. If no charging 12 to 12.5 depending on charge on the battery.
The brush assembly is held in with 2 screws and is easily replaced. Not sure what you are describing as syptoms. Not sure how many brushes you can go through on these, but I'm on my 3rd. Brushes are cheap, alternators are not.
If you have 60 to 100K on the alternator, for sure replace the brushes, but yes it could be the regulator has failed first. Take it to an automotive parts store, most troubleshoot the charging system for free at least around here.