Your argument leaves out necessary steps. It is not a careful analysis, does not consider ways in which it might be mistaken, but gives rise to the impression that you wanted to get to your conclusion as quickly as possible.
There is, necessarily, absolutely no way to determine - given an algorithm - whether it is conscious or not. It is not even a formally undecidable statement!
It is unclear how this follows from anything you wrote.
consciousness refuses to be phrased formally (it is subjective, and computation is objective)
Consider tabooing words like "subjective" and "objective".
we have reduced consciousness to a question about Turing machines - and consciousness refuses to be phrased formally
Similarly, the concept of goodness doesn't make the pursuit of gaining formal control over itself easy on us. Doesn't follow that there is no solution, even less so no worth in pursuit.
In summary, we have reduced existence of consciousness to the existence of a computable process which takes no input and no output - and just is "conscious".
Your concatenation argument leading to this conclusion mixes context and content, obscuring the consciousness abstraction within the whole. There are many obscuring perspectives like this and so they are very weak evidence. You would be more convincing if you can define and attack the consciousness hypothesis directly.
There just remains one detail: There is, necessarily, absolutely no way to determine - given an algorithm - whether it is conscious or not.
This may be a statement of current capability, not a fundamental limit.
... consciousness refuses to be phrased formally (it is subjective, and computation is objective).
Again, this may only be a statement of current capability. Unfortunately you haven't even attempted to provide evidence for this statement.
I have a solution to the problems of other minds, actually. But it requires you to recognize that you are conscious yourself, which is not necessarily possible for all people.
Check: physicalism.com And: qualiacomputing.com/2015/03/31/a-solution-to-the-problem-of-other-minds/
Many rational people are atheists, one does not believe in God for the same reason one does not believe that there is an invisible dragon in my garage. Normally the definition of God would be so vaguely specified that each time you refute it, the bubble pops up elsewhere. Alternatively it may be a triviality, as the definition due to Spinoza.
In Buddhism and Erwin Schrodinger's essay "What is Life?" there is a notion of consciousness - roughly defined as an indivisible subjective experience - some philosophers have even put a currency on it: "qualia". Schrodinger argues that as a consequence of the statistico-determinstic nature of matter as well as the personal experience of directing ones own actions - that one is equal to "omnipresent all-comprehending eternal".
The Church-Turing thesis helps us clarify the situation a little, if we accept it (and we are right to do so, given the current understanding of computation and physics) we must either decide that living beings have some para-physical ability to experience or that there exists some algorithm which (when suitably implemented) becomes conscious. Since the former notion of a para-physical ability is absurd we discharge that.
The algorithm takes as input some stream of bits - we can assume it is also a computable process, call it the environment - processes them and outputs some signals to the environment. Since the concatenation of two computable processes is another computable process we can consider these two processes as one. In summary, we have reduced existence of consciousness to the existence of a computable process which takes no input and no output - and just is "conscious".
There just remains one detail: There is, necessarily, absolutely no way to determine - given an algorithm - whether it is conscious or not. It is not even a formally undecidable statement! Since we have reduced consciousness to a question about Turing machines - and consciousness refuses to be phrased formally (it is subjective, and computation is objective). The notion of consciousness is hence "not even wrong".