What is wrong with this reasoning: The union (finite or infinite) of countable sets is countable. The set of all binary sequences is the infinite union of the sets $S_n$ of all the binary sequences of length n, which are finite, hence countable.
The set of all finite binary sequences is countable, by the argument that you gave in your question. The set of all infinite binary sequences is not countable, by Cantor’s diagonal argument. But the two sets are completely different; indeed, they’re disjoint.