I studied mathematics, and can attest that the attitude of the professor mentioned in the article is very representative of an older generation of mathematicians. Maybe the younger ones are different.
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.