I was quick to judge Bethesda off their buggy past, but it appears they'd already fixed my problem. In either case, be sure to download both the games' Day One patch, update your system manually, and restart the console before playing Fallout 4. Now the elevator works properly - though oddly enough, the music mentioned in the earlier draft of this article never plays.
A new system update downloaded, and rebooted the console. Update November 9th, 5:00PM EST: After fussing with the game for an hour and checking to make sure it was updated on patches, I checked my PlayStation 4 for system updates.
