Elo, what is the advantage of "TheProblem (tm)" over "the problem" or, perhaps better, something like "the biggest problem" or "the problem that should be tackled first" or "the underlying problem or problems"?

It sounds a bit as if you are implicitly proposing a principle like "there is always a single underlying problem, if you can only find it". But that seems rather unlikely to be correct. In any case, whatever you mean it isn't perfectly clear what you do mean, and that is AProblem (tm).