The halting problem is not a problem for constrained sets of programs.
What's up with the huh?
I was commenting on that first line.
I assumed you were thinking of NP, because the halting problem definitely is unsolvable even for constrained sets of programs.
The halting problem itself is completely intractable. When I was at school, Goldbach was the standard example of a program whose halting we can't determine (and which might even be unknowable). These days the Yedidia-Aaronson construction is probably the more common illustration.
I agree that terms like "find all bugs" or "find any bugs" are inherently ambiguous. If halting is considered a bug, then it's trivial to modify a program that halts on a Goldbach counterexample into one that never halts: simply remove the break condition, print the counterexample, and continue.
In my professional work, the systems I work on traverse trillions of states per day and are designed never to revisit one. The only reason they remain analyzable is that their logical structure is far simpler than their state space. In principle, the system should run indefinitely without repeating a state, so - aside from intentional shutdown paths - identifying bugs that cause it to halt is equivalent to solving the halting problem. My hope is that the code is not so convoluted that it becomes an intractable instance of it.
And of course, there are intentional halting states for critical error conditions that "should never happen." Removing those to "solve" the halting problem, as in the Goldbach example above, would itself introduce a bug.