A possible problem might be that a failing area is likely to be undesirable and therefore have low property prices and concomitant taxes. Moving to a more desirable area might be unaffordable. Many people have a strong emotional connection to their home, and would find it difficult to sell it (perhaps at a considerably lower price than they think it is worth) and move somewhere smaller.
Further, there may be people who are "post-economy" in the sense that they have retired. From their point of view, they've paid their dues and have no need to follow the jobs any more. They may have built up local social ties that they don't want to sever by moving.
My point isn't that failing places need to be propped up indefinitely, but that there's a middle ground between "successful, valuable" things and "failing, worthless" things. And there are infrastructure projects (e.g: rural broadband) which can help retain jobs and communities in "failing" places relatively cheaply.