Alright, I finally found the source of the problem.
This on Windows 7 64bit:
If the plugin “Kaleidoscope-HostPowerManagement” is not used and the method HostPowerManagement.enableWakeup() is
not called in the setup() function, the keyboard bugs out: the system hangs for about five seconds, and the keyboard becomes useless.
I tested that with my sketch, and the LEADER plugin example sketch I linked above.