There's got to be more to it than that. The halting problem itself isn't some on-off switch, it can be studied and broken down into different problem classes, some of which are indeed decidable.
It's just impossible to write a general-purpose algorithm to solve it in all cases.
“It's just impossible to write a general-purpose algorithm to solve it in all cases.” That is the definition of an undecidable problem. Independence, as in Euclid’s fifth postulate, is not what we typically think of as undecidability, though can be referred to as such.
It's just impossible to write a general-purpose algorithm to solve it in all cases.