I also studied mathematics, and I can tell you that this was not my experience.

Rather, if a problem is NP-complete/NP-hard it means that we cannot expect a general fast algorithms for exactly this problem (in other words: more mathematics is required, which mathematicians of course love).

But it is absolutely known that there exist other strategies:

- Develop algorithms that work well in practice and make understanding why they work so well in practice your career.

- Find out whether there exists something that makes the instances that occur in practice different from those instances that were used in the proof that the problem is NP-complete/-hard.

- For optimization problems: develop some fast algorithm which guarantees some approximation factor.