Reconnect wireless XBox 360 controller to PC after idle timeout
Solution 1:
It turns out the issue is with Teamspeak. I happened to close TS while the controller was in its "connected to the receiver, but not recognized by Windows" state and that allowed Windows to recognize it. Adding Teamspeak to my searches for the problem led me to the linked thread and the fix.
The issue is apparently something to do with the way Teamspeak's joystick support is broken; disabling or removing the Teamspeak "Gamepad and Joystick Hotkey Support" addon allows the gamepad to reconnect to Windows normally.