Luckily, I had another system at the exact same OS level (this is important), which was working fine. I copied the following apps from that machine to a basic folder on the broken machine:
App Store.app
Terminal.app
Preview.app
TextEdit.app
Of course OS X won't let you copy those apps directly into the Applications folder, so I did this, which recursively copies all of the contents (since the .app files aren't just files) and replaces the existing, non-functioning copies:
cp -R ~/<folder>/<file>.app /Applications/
You may have to really ask for a sandwich:
sudo cp -R ~/<folder>/<file>.app /Applications/
And joy, my system works again (but I haven't deleted the copies yet, just in case).
Hope it helps someone else.