I suppose your argument is, consciousness on the system level does not require individual parts to be conscious.
So let me expand a bit of why I used the adder circuit as an example.
There's nothing about addition that can say "this is addition" in an objective sense. We are dealing with binary encoding of numbers. There's nearly an infinite number of ways to encode information. The design of any circuit deals with particular encoding of information. This extends all the way up to the higher levels. There are an infinite number of ways to encode the state of a go board. Any AI that plays go will deal with some specific ways of encoding the state of the board, and that will be both its input and output.
The most popular "intuitive" model of how consciousness arises is basically hand waving about a system inspecting itself.
But you have to understand, a system inspecting its own state is not any different from a system inspecting any other arbitrary state. For the state of the system will be encoded in some way that was chosen arbitrarily. There's nothing essential about any encoding of any state that can possibly give raise to a phenomenon like consciousness.
You can come study the behavior of any arbitrary system, given certain inputs and outputs, and assign any meaning you want to its states. You can retro-fit any intepretation of state to the system.
Because the thing is with encoding information is it can be completely arbitrary. I can interpret the state [01010101] to be my username, and then interpret [1111000111] to be my real life name. This would allow me to say that a system that takes 01010101 as input and produces 1111000111 is actually a system that can derive my real life name from my username.