A problem being unsolvable within some system does not imply that there is some outer system where it can be solved.

Agreed, I was imprecise before. It is not generally 'a problem' if something is unknown. In the case of the halting problem, it's OK if the algorithm doesn't know when it is going to halt. (This doesn't make it incomplete.) However, it is a problem if X doesn't know how X was created (this makes X incomplete.)

The difference is that an algorithm can be implemeted -- and fully aware of how it is implemented, and know every line of its own code -- without knowing where it is going to halt. Where it's going to halt isn't squirreled away in some other domain to be read at the right moment, the rules for halting are known by the algorithm, it just doesn't know when those rules will be satisfied.

In contrast, X could not have created itself without any source code to do so. The analogous situation would be an algorithm that has halted but doesn't know why it halted. If it cannot know through self-inspection why it halted, then it is incomplete: it must deduce that something outside itself caused it to halt.