That would violate the Church-Turing thesis, which is pretty widely accepted, unless the universe itself is a hypercomputer that we can leverage to perform computation.
But they do, and we can copy them, what is the problem with making a synthetic brain? Why is the natural brain an allowed exception, but the one we make is not allowed?
Edit:
On the one hand you want to use Church-Turing to invalidate the possibility of us creating non-Turing based technology, and yet in a previous post you claimed the human brain likely violates Church-Turing. So which is it? Is reality allowed to violate Church-Turing or is it not?
If you honestly believe that the human brain violates Church-Turing, then you have to explain why you also believe that we can not copy the brain and achieve the same result ourselves.
I do think that I actually agree with your earlier point that once we have a functioning synthetic brain, we can ignore subjective experience itself and just focus on its functional implications for the entire system.
However, there is no reason future computers will be limited to current (Turing) technology instead of being a collection of many co-processors with differing capabilities.
The main point though, is that since consciousness is housed in the physical brain, we can dissect, understand, and recreate fully-functional synthetic versions. While each person is in their own subjective world, the brain which is responsible for it is an objective object that is open to study.