> Actually there's a whole bunch of mathematics which I find useful as an engineer because it tells me that the perfection I have vaguely imagined I could reach for is literally not possible and so I shouldn't expend any effort on that
That's actually given as a reason to study NP-completeness in the classic 1979 book "Computers and Intractability: A Guide to the Theory of NP-Completeness" by Garey & Johnson, which is one of the most cited references in computer science literature.
Chapter one starts with a fictional example. Say you have been trying to develop an algorithm at work that validates designs for new products. After much work you haven't found anything better than exhaustive search, which is too slow.
You don't want to tell your boss "I can't find an efficient algorithm. I guess I'm just too dumb".
What you'd like to do is prove that the problem is inherently intractable, so you could confidently tell your boss "I can't find an efficient algorithm, because no such algorithm is possible!".
Unfortunately, the authors note, proving intractability is also often very hard. Even the best theoreticians have been stymied trying to prove commonly encountered hard problems are intractable. That's where the book comes in:
> However, having read this book, you have discovered something almost as good. The theory of NP-completeness provides many straightforward techniques for proving that a given problem is “just as hard” as a large number of other problems that are widely recognized as being difficult and that have been confounding the experts for years.
Using the techniques from the book you prove the problem is NP-complete. Then you can go to your boss and announce "I can't find an efficient algorithm, but neither can all these famous people". The authors note that at the very least this informs your boss that it won't do any good to fire you and hire another algorithms expert. They go on:
> Of course, our own bosses would frown upon our writing this book if its sole purpose was to protect the jobs of algorithm designers. Indeed, discovering that a problem is NP-complete is usually just the beginning of work on that problem.
...
> However, the knowledge that it is NP-complete does provide valuable information about what lines of approach have the potential of being most productive. Certainly the search for an efficient, exact algorithm should be accorded low priority. It is now more appropriate to concentrate on other, less ambitious, approaches. For example, you might look for efficient algorithms that solve various special cases of the general problem. You might look for algorithms that, though not guaranteed to run quickly, seem likely to do so most of the time. Or you might even relax the problem somewhat, looking for a fast algorithm that merely finds designs that meet most of the component specifications. In short, the primary application of the theory of NP-completeness is to assist algorithm designers in directing their problem-solving efforts toward those approaches that have the greatest likelihood of leading to useful algorithms.