There no fundamental reason why a properly rebuilt alternator wouldn’t perform as well as a new one. So your idea of using a rebuilt unit wasn’t a bad decision. You apparently just got unlucky with the rebuild & testing job on that sample is all. If you suspect you have a bad alternator, many parts stores have a fixture that they’ll use to test it for you as a customer service, free. You have to remove the alternator from the car and take it into the store. As posted above, alternators can be tested on the car too, but sometimes w/that method there can be other problems in the vehicle that can make an alternator look faulty when the alternator isn’t actually the problem. So a parts store alternator test can be very worthwhile.
Another common cause of what seems to be an alternator problem is just that the drive belt is slipping. And with newer cars and their fancy computerized charging management system, that can be where the fault lies too, in the car’s computer system.