I think the idea in the original post was to adjust the goalposts of the original halting problem to get something easier to solve. Instead of looking for programs that "eventually" halt while reproducing the required outputs, one can look for programs that halt "in some reasonable amount of time." The time-bounded halting problem is easier to solve than the original (it is decidable). As one increases the amount of time one views as "reasonable," one gets a sequence of time-bounded halting problems that can be viewed as successively good approximations of the original. There are many similarly "approximate" or "relaxed" versions of the halting problem that are "good enough" in real life and are perfectly decidable.