Unfortunately that was not the solution but I think I have narrowed down the problem (though I am not sure of the solution):
Local Device Found
type: 046d c216
path: sdl://0
serial_number: - 0
Manufacturer:
Product: Logitech Dual Action
Release: 111
Interface: -1
!! Steam controller device opened for index 0.
Couldn't initialize virtual gamepad: Couldn't open /dev/uinput for writing
It looks like I need to give the Steam container write access to the host’s /dev/uinput?