No way would I start by replacing the alternator. I've had an alternator go out once while driving. I used an amazon (Chinese) power cable for a GPS on a road trip once. The mini usb C cable head came off and I heard a pop through the stereo. I kept driving and noticed my battery slowly drain from full 14 volts to completely dead. Near the end, the window motor was killing it, and there wasn't enough juice for the fuel injection. It took right about an hour of driving to kill it. Once your battery is dead it doesn't magically recharge. It needs an alternator to keep it charged or it needs an external connection to supply enough power to get it started. You battery will not die in a mile, and if it were dead, it would not work again because you wish it too. If you are losing all power that quickly, its not a piece going bad, its your electrical system
For the cost of a new alternator, I would either buy a cheap multimeter if you don't have one. Try to test your connections and find an unusual drain by shaking wires, or testing when it happens. You can buy one for less than $20. Otherwise, I would take it to a mechanic, and see if you can pay them for an hour to figure it out. There's a list of things they will know where to start with. If they can't recreate the issue, then they shouldn't charge you to find the problem that's not there. They will know how to check all your connections. Once they figure it out, you can fix it yourself if you want to. An alternator is going to be about the same price as an hour. I know were here because we care about our vehicles, and want to be able to fix everything, but sometimes its just a different pair of eyes, ears, and experience. Not sure what your mechanic/skill level is, not trying to insult you or anyone by saying this.