Not a mathematician, but wouldn't a non-constructive proof would shed some light on the problem so that a constructive proof can be found? In my understanding, computational complexity has a lot of questions we have no idea how to approach, such as P vs NP, and certainly a non-constructive proof -- even of a simpler related problem -- would be quiete handy. Moreover, you seem to have this notion that the only immediate application of computational complexity is cryptography, but this is not necessarily true. Solving problems of computational complexity can help other not-so-related fields of CS such as ML or systems. One immediate reason is that a lot of very fundamental problems are NP Complete, such as graph coloring which appears everywhere in CS such as systems or compilers (e.g. register allocation). Of course most of these problems are probabilistically "solved" with very good heuristics, but this doesn't invalidate further research.
> “wouldn't a non-constructive proof would shed some light on the problem so that a constructive proof can be found?“
This is not necessarily true. Common examples where it doesn’t work that way usually involve proof by contradiction, like in the Brouwer Fixed Point Theorem. The steps of the proof only refute the non-existence of the fixed point, and don’t really offer any constructive insight into how the fixed point can be algorithmically located in general.
Other examples include many elementary calculus theorems, like the intermediate value theorem and Cantor’s diagonalizatiom theorem that the real numbers are uncountable. By extension you can think of the standard proof that the Halting Problem is undecideable as non-constructive.
In complexity theory there are also tricky non-constructive “almost proofs” that are used for intuition a lot.
One example is that it is believed that NP != co-NP. By “contradiction” you can show that if a problem was both NP-complete and also in co-NP, then NP == co-NP. Because of this, when theorists encounter a problem that is in both NP and co-NP, they treat this as “evidence” that the problem is not NP complete.
Overall, sometimes a non-constructive proof offers insight to think of a constructive proof. But often it does not provide any such information.
There is even the branch of logic called intuitionism, which uses the Principle of Explosion to effectively disallow proof by contradiction, which is related to the subset of mathematicians who will only treat fully constructive proofs as acceptable.
