(Answering own question; see my answer below.)
When I tried to run my Today extension in the iOS simulator, Xcode 6 would ask me to choose an app to attach the debugger. After I chose the "Today" app, Xcode would report "Running Today-View on iPhone" but the Today-View would never appear. If I manually pull down the Today-View, my Today extension would not be there.
The funny thing is that, even if I add a brand new app target and a brand new Today extension target to my Xcode project, I still couldn't get Xcode to run and debug a brand new Today extension in the iOS simulator.
I've compared the build settings of my legacy Xcode project against the same of a new Xcode project. There does not seem to be any significant differences between the two.
Why can't Xcode run my Today extension in the iOS simulator against my legacy Xcode project?