After we got home, I did a little bit of research, and read that 1993-98 Tauruses show those symptoms most commonly if the serp belt is slipping around the accessory pulleys. This morning, it started just fine - no problems or funny noises. I took it back to the place that replaced the serp belt anyway and asked them to take a look and see if anything was wonky with it, then left for work. They left me a really terse voicemail message timestamped three minutes after I handed them the keys: "Hi, this is [guy] at [shop]. It's your alternator. *hangup*"
Now, I'm certainly not a mechanic, but I do know that an alternator recharges the battery from the engine power, and that if the alternator wasn't doing that, the battery would drain and the car would stop running and not start again until the battery was charged, rinse and repeat. At no point during all of this was the battery dead. Besides that, when they first replaced the serpentine belt, it took them ten minutes to figure out that the serp belt needed to be replaced, and it was so badly worn that it was cracked like a thirty-year-old rubber band. And now they're telling me that they've not only inspected the serp belt, but also determined that my alternator is broken, in less than five minutes?
I'm definitely getting a second opinion after work. It's sounding to me suspiciously like this guy just wants my car out of his shop, fixed or not.