There are non-computational interpretations of intuitionistic logic too, like every single thing mentioned on this page: https://ncatlab.org/nlab/show/synthetic+mathematics

I think stuff like "synthetic topology", "synthetic differential geometry", "synthetic computability theory", "synthetic algebraic geometry" are the most promising applications at the moment.

It can also find commonalities between different abstract areas of maths, since there are a lot of exotic interpretations of intuitionistic logic, and doing mathematics within intuitionistic logic allows one to prove results which are true in all these interpretations simultaneously.

I'm not sure if intuitionism has a "killer app" yet, but you could say the same about every piece of theory ever, at least over its initial period of development. I think the broad lesson is that the rules of logic are a "coordinate system" for doing mathematics, and changing the rules of logic is like changing to a different coordinate system, which might make certain things easier. In some areas of maths, like modern Algebraic Geometry, the standard rules of logic might be why the area is borderline impenetrable.

These are more like computational-ish interpretations of sheaves, topological spaces, synthetic geometry etc. The link of intuitionistic logic to computation is close enough that these things don't really make it "non-computational". One can definitely argue though that many mathematicians are finding out that things like "expressing X in a topos" are effectively roundabout ways of discussing constructive logic and constructivity concerns.