Well, while I'm certainly not arguing with you, I can only tell of my experience.
I have 3 Explorers currently and 1 is basically a field truck, '97 4door XL 4L OHV - A while back, my cousin and I had a debate on whether the truck would run or not with no battery. We marched out to the truck, fired it up and I took the battery terminals off and the thing didn't even bat an eye, kept running... mind you I didn't have the blower running, lights on or anything. I can't tell you how long it ran for because I can't remember how much time elapsed before putting the terminals back on, I just remember that it didn't stall. (Which proved me wrong, because I had declared that it wouldn't run without the battery, lol)
In the OP's case, I would put the good battery in, fire the truck up.. let it run for a few minutes and then test the voltage with a multimeter... doing this of course with all electronics/blower/lights off. If the gauge in the truck read in the middle and the voltmeter(multimeter) read 14ish, I would leave the alternator alone and call it a day.