Have you already checked if there's any weird noise in the engine when it starts, like a squealing sound? Because besides what they told you about the volts (and they're totally right that 13V is way too low), there's another pretty common culprit that people forget about: the alternator belt can be worn out or loose, and even if the alternator itself is fine, it's not spinning enough to charge. If the belt is broken or really slack, the alternator doesn't work, and your battery drains even though it's technically connected.
Before you spend money on a new alternator, what you can do is visually check that belt (pop the hood, look at the front of the engine) and see if it has cracks, is frayed, or looks really dirty. If it's loose, you'll probably see it moving a ton when you touch it carefully. You can also have someone start the engine while you watch to see if the belt is spinning smoothly without slipping. The belt is way cheaper than replacing the whole alternator 🚗
If the belt looks normal, then it's probably the alternator itself. But at least that way you rule out something that's relatively easy to check first.